Matricial Wasserstein-1 Distance
Yongxin Chen Tryphon T. Georgiou Lipeng Ning Allen Tannenbaum





Matricial Wasserstein-1 Distance
    Yongxin Chen, Tryphon T. Georgiou, Lipeng Ning,  and Allen Tannenbaum
Y. Chen is with the Department of Medical Physics, Memorial Sloan Kettering Cancer Center, NY; email: chen2468@umn.edu
T. T. Georgiou is with the Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA; email: tryphon@uci.edu
L. Ning is with Brigham and Women's Hospital (Harvard Medical School), MA.; email: lning@bwh.harvard.edu
A. Tannenbaum is with the Departments of Computer Science and Applied Mathematics & Statistics, Stony Brook University, NY; email: allen.tannenbaum@stonybrook.edu
    December 30, 2023
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We propose an extension of the Wasserstein 1-metric (W_1) for matrix probability densities, matrix-valued density measures, and an unbalanced interpretation of mass transport. We use duality theory and, in particular, a “dual of the dual” formulation of W_1. This matrix analogue of the Earth Mover's Distance has several attractive features including ease of computation.




§ INTRODUCTION


Optimal mass transport (OMT) has proven to be a powerful methodology for numerous problems in physics, probability, information theory, fluid mechanics, econometrics, systems and control, computer vision, and signal/image processing <cit.>.
Developments along purely controls-related issues ensued when it was recognized that mass transport may be naturally reformulated as a stochastic control problem; see <cit.> and the references therein.

Historically, the problem of OMT <cit.> began with the question of minimizing the effort of transporting one distribution to another, typically with a cost proportional to the Euclidean distance between starting and ending points of the mass being transported. However, the control-theoretic reformulation <cit.> which was at the root of the aforementioned developments was based on the choice of a quadratic cost. The quadratic cost allowed the interpretation of the transport effort as an action integral and gave rise to a Riemannian structure on the space of distributions <cit.>. The originality in our present work is two-fold. First, we formulate the  transport problem with an L_1 cost in a similar manner, as a control problem with an L_1-path cost functional, and secondly,  we develop theory for shaping flows of matrix-valued distributions which is a non-trivial generalization of classical OMT.

The relevance of OMT on flows of matrix-valued distributions was already recognized in <cit.> and was cast as a control problem as well, albeit in a quadratic-cost setting. At that point, interest in the geometry of matrix-valued distributions stemmed from applications to spectral analysis of vector-valued time series (see <cit.> and the references therein). Yet soon it became aparent that flows of matrix-valued distributions represent evolution of quantum systems. In fact, there has been a burst of activity in applying ideas of quantum mechanics to OMT of matrix-valued densities as well as, utilizing an OMT framework to study the dynamics of quantum systems: three groups <cit.> independently and simultaneously developed quantum mechanical frameworks for defining a Wasserstein-2 distance on matrix-valued densities (normalized to have trace 1), via a variational formalism generalizing the  work of <cit.>. We note that <cit.> develop matrix-valued generalizations of the Wasserstein 2-metric (W_2) and explore the Riemannian-like structure for studying the entropic flows of quantum states.

Thus, in our present note, we develop a natural extension of the Wasserstein 1-metric to matrix-valued densities and matrix-valued measures.
Our point of view is somewhat different from the earlier works on matricial Wassserstein-2 metrics. We mainly use duality theory <cit.>. Further, we do not employ the Benamou and Brenier <cit.> control formulation of OMT, but rather the Kantorovich-Rubinstein duality. This new scheme is computationally more attractive and, moreover, it is especially appealing when specialized to weighted graphs (discrete spaces) that are sparse (few edges), as is the case for many real-world networks <cit.>.

The present paper is structured as follows. Section <ref> is a quick review of several different formulations of Wasserstein-1 distance in the scalar setting. Using the quantum gradient operator defined in Section <ref>, we generalize the Wasserstein-1 metric to the space of density matrices in Section <ref>. The case where the two marginal matrices have different traces is discussed in Section <ref>. We finally extend the framework to deal with matrix-valued densities in Section <ref>, which may find applications in multivariate spectra analysis as well as comparing stable multi-inputs multi-outputs (MIMO) systems. The paper concludes with an academic example in Section <ref>.




§ OPTIMAL MASS TRANSPORT

We begin with duality theory, explained for scalar densities, upon which our matricial generalization of the Wasserstein-1 metric is based.

Given two probability densities ρ_0 and ρ_1 on ^m, the Wasserstein-1 distance between them is
	
    W_1 (ρ_0,ρ_1):=inf_π∈Π(ρ_0,ρ_1)∫_^m×^mx-yπ(dx,dy),

where Π(ρ_0,ρ_1) denotes the set of couplings between ρ_0 and ρ_1. The Wasserstein-1 distance has a dual formulation via the following result due to Kantorovich and Rubinstein <cit.>:
	
    W_1 (ρ_0,ρ_1)   =sup_f {∫_^m f(x)(ρ_0(x)-ρ_1(x))dx.
       .∫_^mf_ Lip≤ 1},

	where f_ Lip denotes the Lipschitz constant.
When f is differentiable, f_ Lip =∇_x f. It follows that,
	
    W_1 (ρ_0,ρ_1)   =sup_f {∫_^m f(x)(ρ_0(x)-ρ_1(x))dx.
       .∫_^m∇_x f≤ 1},
	
Starting from (<ref>), by once again considering the dual, we readily obtain the very important reformulation
	
    W_1 (ρ_0,ρ_1)   =inf_u(·){∫_^mu(x) dx .
       .∫_^mρ_0-ρ_1 +∇_x· u=0 },

where the (Lagrange) optimization variable u now represents flux. Alternatively, this can be written as the control-optimization problem in the Benamou-Brenier style <cit.>
	
    W_1 (ρ_0,ρ_1)   =inf_u(·,·){∫_0^1∫_^mu(t,x) dx dt.
       x∫_^m∂ρ(t,x)/∂ t+∇_x· u(t,x)=0,
       .∫_^m ρ(0,x)=ρ_0(x), ρ_1(1,x)=ρ_1(x) },


This “dual of the dual” formulation turns the Kantorovich and Rubinstein into a control problem to determine a suitable velocity (control vector) u. We remark that from a computational standpoint, when applied to discrete spaces (graphs),
this formulation leads to a very substantial computational benefit in the case of sparse graphs; this is due to the fact that (<ref>)
involves solving systems of the order of the square of the number of nodes, while equation (<ref>), solving systems of the order of the number of edges.




§ GRADIENT ON SPACE OF HERMITIAN MATRICES


We closely follow the treatment in <cit.>. In particular, we will need a notion of gradient on the space of Hermitian matrices and its dual, i.e. the divergence.

Denote by  and  the set of  n× n Hermitian and skew-Hermitian matrices, respectively. We will assume that all of our matrices are of fixed size n× n. Next, we denote the space of block-column vectors consisting of N elements in  and  as ^N and ^N, respectively. We also let _+ and _++ denote the cones of nonnegative and positive-definite matrices, respectively, and
	
    :=   {ρ∈_+|(ρ)=1},
    _+    :=   {ρ∈_++|(ρ)=1}.

We note that the tangent space of _+, at any ρ∈+ is given by
	
    T_ρ={δ∈|(δ)=0},

and we use the standard notion of inner product, namely
	
    ⟨ X,Y⟩=(X^*Y),

for both  and .
For X, Y∈^N (^N),
	
    ⟨ X, Y⟩=∑_k=1^N (X_k^*Y_k).

Given X=[X_1^*,⋯,X_N^*]^* ∈^N (^N), Y∈ (), set
	
    XY=[[ X_1;   ⋮; X_N ]]Y
    		:=
    		[[  X_1Y;     ⋮; X_N Y ]],

and
	
    YX=Y[[ X_1;   ⋮; X_N ]]
    		:=
    		[[ YX_1;    ⋮; YX_N ]].



For a given L∈^N we define
	
    ∇_L: →^N,   X ↦[ [  L_1 X-XL_1;           ⋮; L_N X-X L_N ]]

to be the gradient operator.
By analogy with the ordinary multivariable calculus, we refer to its dual with respect to the Hilbert-Schmidt inner product as the (negative) divergence operator, and this is
	
    ∇_L^*: ^N →,  Y=
    		[ [ Y_1;   ⋮; Y_N ]]
    		↦∑_k^N L_k Y_k-Y_k L_k,

i.e., ∇_L^* is defined by means of the identity
	
    ⟨∇_L X , Y⟩ =⟨ X , ∇_L^* Y⟩.

A standing assumption throughout, is that the null space of ∇_L, denoted by ker(∇_L), contains only scalar multiples of the identity matrix.



§ WASSERTEIN-1 DISTANCE FOR DENSITY MATRICES


In this section, we show that both (<ref>) and (<ref>) have natural counterparts for probability density matrices, i.e. matrices in .
This set-up obviously works for matrices in _+ of equal trace.

We treat (<ref>) as our starting definition and define the W_1 distance in the space of density matrices as
	
    W_1 (ρ_0,ρ_1):=sup_f∈{[f(ρ_0-ρ_1)] | ∇_L f≤ 1}.

Here · is the operator norm. The above is well-defined since by assumption, the null space of ∇_L is spanned by the identity matrix I. As above, we have that
	
    ∇_L f=
    		[[  L_1f-fL_1;          ⋮; L_N f-fL_N ]].

This should be compared to the Connes spectral distance <cit.>, which is given by
	
    d_D(ρ_0,ρ_1)=sup_f∈{[f(ρ_0-ρ_1)] | [D,f]≤ 1}.

	
It is not difficult to see that the dual of (<ref>) is
	
    Ŵ_1 (ρ_0,ρ_1)=inf_u∈^N{u_* | ρ_0-ρ_1-∇_L^* u=0},

which is the counterpart of (<ref>). Here ·_* denotes the nuclear norm <cit.>.
In particular, we have the following theorems.

 Notation as above. Then
	
    W_1 (ρ_0,ρ_1)=Ŵ_1 (ρ_0,ρ_1).




We start from (<ref>) and use the fact that
	
    u_*=sup_g∈^N, g≤ 1⟨ u, g⟩.

It follows
	
    Ŵ_1 (ρ_0,ρ_1)   =   inf_usup_f{u_*
    	+⟨ f, ρ_0-ρ_1-∇_L^* u⟩}
       =   inf_usup_f,g≤ 1{⟨ u, g⟩
    	+⟨ f, ρ_0-ρ_1-∇_L^* u⟩}
       =   inf_usup_f,g≤ 1{⟨ u, g-∇_Lf⟩
    	+⟨ f, ρ_0-ρ_1⟩}
       ≥   sup_f,g≤ 1inf_u{⟨ u, g-∇_Lf⟩
    	+⟨ f, ρ_0-ρ_1⟩}
       =   sup_f,g≤ 1{⟨ f, ρ_0-ρ_1⟩ | 
    	g=∇_Lf}
       =   sup_f{⟨ f, ρ_0-ρ_1⟩ | ∇_L f≤ 1}
       =    W_1(ρ_0,ρ_1).

This implies that (<ref>) and (<ref>) are dual to each other. Since both of them are strictly feasible, the duality gap is zero. Therefore W_1 (ρ_0,ρ_1)=Ŵ_1 (ρ_0,ρ_1).



The W_1 distance defined as in (<ref>) is a metric on the space of density matrices .


Obviously W_1(ρ_0,ρ_1)≥ 0 holds with equality if and only if ρ_0=ρ_1. The symmetric property that W_1(ρ_0,ρ_1)=W_1(ρ_1,ρ_0) is also clear from the definition. Here we prove the triangle inequality. That is, for any ρ_0, ρ_1, ρ_2∈), we have
	
    W_1(ρ_0,ρ_2)≤ W_1(ρ_0,ρ_1)+W_1(ρ_1,ρ_2).

It is easier to see this from the dual formulation (<ref>). Let u_1, u_2 be the optimal fluxes for (ρ_0,ρ_1) and (ρ_1,ρ_2) respectively. Then u_1+u_2 is a feasible flux for (ρ_0,ρ_2), namely,
	
    ρ_0-ρ_2-∇_L^* (u_1+u_2)=0.

It follows that
	
    W_1(ρ_0,ρ_2)≤u_1+u_2_*≤u_1_*+u_2_*,

which completes the proof.




§ WASSERTEIN-1 DISTANCE: THE UNBALANCED CASE

In this section, we extend the definition of Wasserstein-1 distance to the space nonnegative matrices _+, i.e., we remove the constraint of both matrices having equal traces. Compare also with some very interesting recent  work <cit.> on fast computational methods for W_1 in the unbalanced scalar case.

In order to compare matrices of unequal trace we relax the constraint in (<ref>), which forces
	(ρ_0)=(ρ_1),
by introducing a “source” term v∈. That is, we replace our continuity equation (<ref>) with
	
    ρ_0-ρ_1-∇_L^* u-v=0.

With this added source, we define a Wasserstein-1 distance in _+ as follows.   Given ρ_0, ρ_1∈_+, we define
	
    V_1 (ρ_0,ρ_1)=inf_u∈^N
    v∈{u_*+αv_*|ρ_0-ρ_1-∇_L^* u-v=0}.

Here α>0 measures the relative significance between u and v.

Another natural way to compare ρ_0,ρ_1∈_+ is by finding μ, ν∈_+ having equal trace that are close to ρ_0, ρ_1 in some norm (here taken to be the nuclear norm), as well as close to one another. More specifically, we seek μ, ν to minimize
	
    W_1(μ,ν)+αρ_0-μ_*+αρ_1-ν_*.

Putting the two terms together we obtain the following definition of Wasserstein-1 distance
	
	
    V̂_1(ρ_0,ρ_1)   =   inf_u∈^N 
    μ, ν∈_+u_*+αρ_0-μ_*+αρ_1-ν_*
    	
          μ-ν-∇_L^* u=0,
    	
          (μ)=(ν).

	
It turns out these two relaxations of W_1 are in fact equivalent.


 With notation and assumptions as above,
	
    V_1(ρ_0,ρ_1)= V̂_1(ρ_0,ρ_1).



Clearly, V̂_1(ρ_0,ρ_1) ≥ V_1(ρ_0,ρ_1). On the other hand, let u, v be a minimizer of (<ref>), and v=v_1-v_0 with v_0,v_1∈_+, i.e., v_0,v_1 are the negative and positive parts of v respectively, then μ=ρ_0+v_0, ν=ρ_1+v_1 together with u is a feasible solution to (<ref>). With this solution,
	
    V̂_1(ρ_0,ρ_1)    ≤   u_*+αρ_0-μ_*+αρ_1-ν_*
    		
       =   u_*+αv_0_*+αv_1_*
    		
       =   u_*+αv_*,

which implies that V̂_1(ρ_0,ρ_1)≤ V_1(ρ_0,ρ_1). This completes the proof.



The formula (<ref>) defines a metric on _+.


The proof follows exactly the same lines as in Theorem <ref>.


Using the technique of Lagrangian multipliers one can deduce the dual formulation of (<ref>) and establish the following:
 Notation as above. Then
	
    V_1(ρ_0,ρ_1)=sup_f∈{[f(ρ_0-ρ_1)] | ∇_L f≤ 1,  f≤α}.



Straight calculation gives
	
    V_1 (ρ_0,ρ_1)   =   inf_u,vsup_f{u_*+αv_*+
    	
          ⟨ f, ρ_0-ρ_1-∇_L^* u-v⟩}
       =   inf_u,vsup_f,g≤ 1, h≤ 1{⟨ u, g⟩+α⟨ v,h⟩
          +
    	⟨ f, ρ_0-ρ_1-∇_L^* u-v⟩}
          ≥sup_f,g≤ 1, h≤ 1inf_u,v{⟨ u, g-∇_L f⟩
    	+⟨ v,α h-f⟩
          +
    	⟨ f, ρ_0-ρ_1⟩}
          =sup_f {⟨ f, ρ_0-ρ_1⟩ | ∇_L f≤ 1,  f≤α}.

This together with the strong duality completes the proof.





§ WASSERSTEIN-1 DISTANCE FOR MATRIX-VALUED DENSITIES

With little effort we are able to generalize the definition of Wasserstein-1 distance to the space of matrix-valued densities. Examples of matrix-valued densities include power spectra of multivariate time series, stress tensors, diffusion tensors and so on, and hence our motivation in considering matrix-valued distribution on possibly more than a one dimensional spatial coordinates.

Given two matrix-valued densities ρ_0, ρ_1 satisfying
	
    ∫_^m(ρ_0(x))dx=∫_^m(ρ_1(x))dx,

 we can define their Wasserstein-1 distance as
	
    W_1 (ρ_0,ρ_1)   :=   sup_f∈{∫_^m[f(x)(ρ_0(x)-ρ_1(x))]dx | .
    		
          .[∇_x f
    ∇_L f]≤ 1},

or through its dual
	
    W_1 (ρ_0,ρ_1)   =   inf_u_1∈^m 
     u_2∈^N{∫_^m[u_1(x)
    u_2(x)]_*dx | .
    		
          .
    		ρ_0-ρ_1+∇_x· u_1-∇_L^* u_2=0}.


For more general densities where condition (<ref>) may not be valid, we define
	 
    V_1 (ρ_0,ρ_1)   :=   sup_f∈{∫_^m[f(x)(ρ_0(x)-ρ_1(x))]dx | . 
          .
    		[∇_x f
    ∇_L f]≤ 1, f≤α},

or, equivalently,
	
    V_1 (ρ_0,ρ_1)   =   inf_u_1∈^m 
     u_2∈^N, v∈{∫_^m[u_1(x)
    u_2(x)]_*+αv_*dx | .
          .
    		ρ_0-ρ_1+∇_x· u_1-∇_L^* u_2-v=0}.


One can introduce positive coefficients β_1>0, β_2>0 to  trade-off the relative importance of u_1 and u_2 in establishing correspondence between the two distributions as follows:
	 
    V_1 (ρ_0,ρ_1)   :=sup_f∈{∫_^m[f(x)(ρ_0(x)-ρ_1(x))]dx | .
       .
    		[β_1∇_x f
    β_2∇_L f]≤ 1, f≤α},

or, equivalently,
	
    V_1 (ρ_0,ρ_1)   =inf_u_1∈^m 
     u_2∈^N, v∈{∫_^m[u_1(x)
    u_2(x))_*+αv_*dx | .
       .
    		ρ_0-ρ_1+β_1∇_x· u_1-β_2∇_L^* u_2-v=0}.

	


§ EXAMPLE

We use our framework to compare power spectra of multivariate time series (in discrete time). Evidently, the distance between two power spectra induces a distance between corresponding linear modeling filters and, thereby, can be used to compare (stable) MIMO systems <cit.>.

Consider the three power spectra as shown in Figure <ref> (in different colors). What is shown in the three subplots are power spectra of two time series (in subplots (a) and (c)) and their cross-spectrum (in subplot (b)) as functions of time (the phase of the cross spectra are not shown). Thus, the three different colors represent the three different matrix-valued power spectra given by:
	
    ρ_0(θ)    = [1     0.4 
    0     1]
    		[0.01     0 
    0    0.7/|a_0(e^jθ)|^2]
    		[1     0 
    0.4     1]
    		
    ρ_1(θ)    =[1     0.5 
    0.5e^jθ    1][0.5/|a_1(e^jθ)|^2    0 
     0    0.5/|a_1(e^jθ)|^2]
    		[1     0.5e^-jθ
    0.5     1]
    ρ_2(θ)    = [1     0 
    0.4e^jθ    1]
    		[2/|a_2(e^jθ)|^2    0 
    0     0.02 ]
    		[1     0.4e^-jθ
     0     1]

where
	
    a_0(z)   =(1-1.9cos(π/6)z-0.95^2z^2)
       ×(1-1.5cos(π/3)z+0.75^2z^2)
    	
    
    	a_1(z)   =(1-1.9cos(2π/3)z-0.95^2z^2)
       ×(1-1.5cos(5π/8)z+0.75^2z^2)
    	
    
    	a_2(z)   = (1-1.9cos(5π/12)z-0.95^2z^2)
       ×(1-1.5cos(π/2)z+0.75^2z^2).




The distances between the each pair for different β_1,β_2 values, α=1, and the choice L=[L_1,  L_2] with

    L1=[1    0
    0      0], L_2=[1    1
    1      0],
 are tabulated in Table <ref>. We observe that when the penalty on the rotation part is large (β_1>>β_2),  we have V_1(ρ_0,ρ_2)>V_1(ρ_0,ρ_1) and V_1(ρ_0,ρ_2)>V_1(ρ_2,ρ_1). On the other hand, when the penalty on translation is large relative to the cost of rotation (β_1<<β_2), we have V_1(ρ_0,ρ_1)>V_1(ρ_0,ρ_2) and V_1(ρ_0,ρ_1)>V_1(ρ_1,ρ_2). These findings are in agreement with the intuition when observing the relative frequency directionality of power in the three spectra. More specifically, ρ_1 requires a significant drift in directionality before we can match it with the other two, while this is less important when comparing ρ_0 and ρ_2. For this latter case, it is the actual frequency where the power resides that distinguishes the two while the directionality is more in agreement.

What this example underscores is the ability of the metric to be tailored to applications where we need to trade off and compromise, in a principled way, between two vastly different features of matrix-valued distributions, i.e., spatial location versus directionality of the “intensity.” What was achieved in this paper is the construction of a suitable and easily computable metric that can be utilized for this purpose.





§ FUTURE RESEARCH


We introduced generalization of the scalar W_1 distance to matrices and matrix-valued measures. This new metric, W_1, is computationally simpler and more attractive than earlier metrics, based on quadratic cost criteria. In fact, our “dual of the dual” formulation makes the metric especially attractive when comparing matrix-valued data on a discrete space (graph, network).

We note that the Wasserstein 1-metric has been used as a tool in defining curvature <cit.> and in analyzing the robustness of complex networks derived from scalar-valued data <cit.>. The formalism presented in the current work, suggests alternative notions of curvature and robustness when the nodes of a network carry matrix-valued data, e.g., in diffusion tensor imaging. We plan to pursue such issues in future work.



§ ACKNOWLEDGEMENTS

This project was supported by AFOSR grants (FA9550-15-1-0045 and FA9550-17-1-0435), grants from the National Center for Research Resources (P41-
RR-013218) and the National Institute of Biomedical Imaging and Bioengineering (P41-EB-015902), National Science Foundation (NSF), and National Institutes of Health (P30-CA-008748 and 1U24CA18092401A1).

plain
99






French
J.-D.  Benamou and Y. Brenier,
“A computational fluid mechanics solution to the
Monge–Kantorovich mass transfer problem,”
Numerische Mathematik 84 (2000), pp. 375-393.

nuclear E. Candes and T. Tao, “The power of convex relaxation: Near- 
optimal matrix completion,” IEEE Trans. Inform. Theory 56:5 (2009), pp. 2053-2080.

Carlen
E. Carlen and J. Maas, “Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance,” https://arxiv.org/abs/1609.01254, 2016.

Chen Y. Chen, T. T. Georgiou, A. Tannenbaum, “Matrix optimal mass transport: a quantum mechanical approach,” https://arxiv.org/abs/1610.03041, 2016.

CGP Y. Chen, T. T. Georgiou, M. Pavon, “On the relation between optimal transport and Schrödinger bridges: A stochastic control viewpoint,” Journal of Optimization Theory and Applications 169:2 (2016), pp. 671-691.

CGP0 Y. Chen, T. T. Georgiou, M. Pavon, “Optimal transport over a linear dynamical system,” IEEE Transactions on Automatic Control, to appear.

CGP1 Y. Chen, T. T. Georgiou, M. Pavon, “Optimal steering of a linear stochastic system to a final probability distribution, Part I,” IEEE Transactions on Automatic Control 61:5 (2016), pp. 1158-1169.

CGP2 Y. Chen, T. T. Georgiou, M. Pavon, “Optimal steering of a linear stochastic system to a final probability distribution, Part II,”   IEEE Transactions on Automatic Control 61:5 (2016), pp. 1170-1180.

CGPT  Y. Chen, T. T. Georgiou, M. Pavon, and A. Tannenbaum, “Robust transport over networks,” IEEE Transactions on Automatic Control, to appear.




connes ] A. Connes, Noncommutative Geometry, Academic Press Inc., San Diego (1994), available at
http://www.alainconnes.org/downloads.html.

DP P. Dai Pra, “A stochastic control approach to reciprocal diffusion processes,” Applied Mathematics and Optimization 23:1 (1991), pp. 313-329.

Evans
L. C. Evans, Partial differential equations and
Monge–Kantorovich mass transfer, in Current Developments in
Mathematics, International Press, Boston, MA, 1999, pp. 65–126.






Jordan
R. Jordan, D. Kinderlehrer, and F. Otto, “The variational formulation of the Fokker-Planck equation,”
SIAM J. Math. Anal. 29 (1998), pp. 1-17.

Kantorovich1948
L. V. Kantorovich, “On a problem of Monge,” Uspekhi Mat.
Nauk. 3 (1948), pp. 225–226.




PW M. Pavon and A. Wakolbinger, “On free energy, stochastic control, and Schrödinger processes,” Modeling, Estimation and Control of Systems with Uncertainty, G.B. Di Masi, A.Gombani, A.Kurzhanski Eds., Birkauser, Boston, pp. 334-348 (1991).

Leonard C. Léonard, “A survey of the Schrödinger problem and some of its connections with optimal transport,” Discrete Contin. Dyn. Syst. A 34(4) (2014), pp. 1533-1574.

osher W. Li, P. Yin, and S. Osher, “A fast algorithm for unbalanced L^1 Monge-Kantorovich problem,” preprint.


McC97 R. McCann, “A convexity principle for interacting gases,” Adv. Math. 128 (1997),
pp. 153–179.


Mielke M. Mittnenzweig and A. Mielke, “An entropic gradient structure for Lindblad equations and GENERIC for quantum systems coupled to macroscopic models,” https://arxiv.org/abs/1609.05765, 2016.

Mueller M  Mueller, P. Karasev, I. Kolesov, and A. Tannenbaum, “Optical flow estimation for
flame detection in videos,” IEEE Trans. Image Processing 22:7 (2013), pp. 2786-2797.

MT T. Mikami and M. Thieullen, “Optimal transportation problem by stochastic optimal control,” SIAM Journal Control and Optimization 47:3 (2008), pp. 1127-1139.

Lipeng
L. Ning, T. Georgiou, and A. Tannenbaum, “On matrix–valued Monge-Kantorovich optimal mass transport,” IEEE Transactions on Automatic Control 60:2 (2015), pp. 373-382.

NinGeo14
L. Ning, and Tryphon T. Georgiou, “Metrics for matrix-valued measures via test functions,” 53rd IEEE Conference on Decision and Control (CDC), 2014.

Ollivier Y. Olliver, “Ricci curvature of Markov chains on metric spaces,” J. Funct. Anal. 256 (2009), pp. 810-864.

Otto
F. Otto, “The geometry of dissipative evolution equations: the porous medium equation,” Communications in Partial Differential Equations 26 (2001), pp. 101-174.

Rachev
S. Rachev and L. Rüschendorf,
Mass Transportation Problems, Volumes I and II, Probability
and Its Applications, Springer, New York, 1998.

Sandhu R.Sandhu, T. Georgiou, E. Reznik, L. Zhu, I. Kolesov,
Y. Senbabaoglu, and A. Tannenbaum1, “Graph curvature for differentiating cancer networks,” Scientific Reports (Nature), vol. 5, 12323; doi: 10.1038/srep12323 (2015).

Sandhu1 R. Sandhu, T. Georgiou, and A. Tannenbaum, “Ricci curvature: An economic indicator for market
fragility and systemic risk,” Science Advances, vol. 2, doi: 10.1126/sciadv.1501495, 2016.

TGT E. Tannenbaum, T. Georgiou, and A. Tannenbaum, A., “Signals and control aspects of optimal mass transport and the Boltzmann entropy,” in 49th IEEE Conference on Decision and Control (CDC), December 2010.



Villani C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58, AMS,
Providence, RI, 2003.

jonck C.Wang, E. Jonckheere, and R. Banirazi, “Wireless network capacity versus Ollivier-Ricci curvature under Heat
Diffusion (HD) protocol,” Proceedings of ACC, 2013.



