A note on flatness of non separable tangent cone
Thibaut Le Gouic

A Key 6G Challenge and Opportunity - Connecting the Remaining 4 Billions: A Survey on Rural Connectivity
    Elias Yaacoub^1, Senior Member, IEEE,
        and Mohamed-Slim Alouini^2, Fellow, IEEE

^1Department
of Electrical and Computer Engineering, American University of Beirut, P.O. Box 11-0236 / ECE Department, Riad El-Solh / Beirut 1107 2020,
Lebanon. E-mail: eliasy@ieee.org.
^2Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Makkah Province, Saudi Arabia. E-mail: slim.alouini@kaust.edu.sa.

    December 30, 2023
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Given a probability measure  on an Alexandrov space  with curvature bounded below, we prove that the support of the pushforward of  on the tangent cone T_ at its (exponential) barycenter  is a subset of a Hilbert space, without separability of the tangent cone.




§ INTRODUCTION

Barycenter of a probability measure  (a.k.a. Fréchet means) provides an extension of expectation on Euclidean space to arbitrary metric spaces.
We present here a useful tool for the study of barycenters on Alexandrov spaces with curvature bounded below: the support of log_# in the tangent cone at the barycenter is included in a Hilbert space.
This result has been stated in <cit.> as Theorem 45, however the proof is not written.
Moreover, there is an extra assumption of support of log_# being separable, which does not even seem to be a consequence of the support of  being separable.
This paper present a proof of this result, without this extra separable assumption.
The proof is essentially the one of Theorem 45 of <cit.>, with needed approximations dealt with a bit differently.




§ SETTING AND MAIN RESULT


We use a classical notion of curvature bounded below for geodesic spaces, referred to as Alexandrov curvature.
We recall several notions whose formal definitions can be found for instance in <cit.> or in the work in progress <cit.>.

For a metric space (,d), we denote by  that set of probability measures  on  with finite moment of order 2 (i.e. there exists x∈ such that ∫ d^2(x,y)d(y)<∞).
The support of a measure  will be denoted by .

A geodesic space is a metric space (,d) such that every two points x,y∈ at distance is connected by a curve of length d(x,y).
Such shortest curves are called geodesics.
For κ∈, the model space (_κ,d_κ) denotes the 2-dimensional surface of constant Gauss curvature κ.
A geodesic space (,d) is an Alexandrov space with curvature bounded below by κ∈ if for every triangle (3-uple) (x_0,x_1,y)∈, and a constant speed geodesic (x_t)_t∈ [0;1] there exists an isometric triangle (x̃_0,x̃_1,ỹ)∈_κ, such that the geodesic (x̃_t)_t∈[0;1] satisfies for all t∈ [0;1],

    d(y,x_t)≥ d(ỹ,x̃_t).

For such spaces, angles between two unit-speed geodesics γ_1,γ_2 starting at the same point p∈ can be defined as follows:

    cos∠_p(γ_1,γ_2)=lim_t→ 0d^2(γ_1(t),p)+d^2(γ_2(t),p)-d^2(γ_1(t),γ_2(t))/2d(p,γ_1(t))d(p,γ_2(t)),

where angle ∠_p(γ_1,γ_2)∈[0;π].
Denote by Γ_p the set of all unit-speed geodesics emanating from x.
Using angles, we can define the tangent cone T_p at p∈ as follows.
First define T'_p as the (quotient) set Γ_x×^+, equipped with the (pseudo-)metric defined by

    (γ_1,t)-(γ_2,s)^2_p := s^2 + t^2 - 2s.tcos∠_p(γ_1,γ_2).

Then, the tangent cone T_p is defined as the completion of T_p.
We will use the notation for u,v∈ T_p,

    ⟨ u,v⟩_p:=1/2(u^2+v^2-u-v^2),

We will often identify a point γ(t)∈ with (γ,t)∈ T_p.
Although such γ might not be unique, we will implicitly assume the choice of a measurable map log_p:→ T_p, called logarithmic map, such that for all x∈, there exists a geodesic γ emanating from p such that, for some t>0, γ(t)=x and

    log_p(x)=(γ,t).

Then the pushforward of  by log_p will be denoted by #log_p.

The tangent cone is not necessarily a geodesic space (see <cit.>), however, it is included in a geodesic space - namely the ultratangent space (see for instance Theorem 14.4.2 and 14.4.1 of <cit.>) that is an Alexandrov space with curvature bounded below by 0.

The tangent cone T_p contains the subspace _p of all points with an opposite, formally defined as follows.
A point u belongs to _p⊂ T_p if and only if there exists v∈ T_p such that u_p=v_p and

    ⟨ u,v⟩_p = -u_p^2.

Our main result is based on the following Theorem.
[Theorem 14.5.4 in <cit.>]
The set _p equipped with the induced metric of T_p is a Hilbert space.


A point ∈ is a barycenter of the probability measure  if for all b∈

    ∫ d^2(x,)d(x)≤∫ d^2(x,b)d(x).

Such barycenter might not be unique, neither exist.
However, when they exist, they satisfy

    ∫⟨ x,y⟩_ d⊗(x,y)=0.

A point ∈ satisfying (<ref>) is called an exponential barycenter of .

We can now state our main result.


Let (,d) be an Alexandrov space with curvature bounded below by some κ∈ and ∈.
If ∈ is an exponential barycenter of , then the (log_#)⊂_.
In particular, (log_#) is included in a Hilbert space.


This result allows to prove the following Corollary, that has been implicitly used in <cit.>.



Let b∈ T_
Then, the map ⟨ .,b⟩_:_→ is continuous and linear.
In particular, if  is an exponential barycenter of , then

    ∫⟨ x,b⟩_d(x)=0.






§ PROOFS

Recall that we always identify a point in  and its image in the tangent cone T_p by the log_p map.


We check that x↦⟨ x,b⟩_ is a convex and concave function in _.
Let t∈ (0,1), x_0,x_1 in _, and set x_t=(1-t)x_0+tx_1.
Since the tangent cone is included in an Alexandrov space with curvature bounded below by 0 on one hand, and _ is a Hilbert space on the other hand,

    ⟨ x_t,b⟩_   =1/2(x_t_^2+b_^2-x_t-b^2)
       ≤1/2((1-t)(x_0_^2-x_0-b_^2)+t(x_1_^2-x_1-b^2)+b_^2)
       =(1-t)⟨ x_0,b⟩_+t⟨ x_1,b⟩_.

The same lines applied to -x_0 and -x_1 gives the converse inequality

    ⟨ -x_t,b⟩_≤ (1-t)⟨ -x_0,b⟩_ + t⟨ -x_1,b⟩_.

The second statement follows from the fact that  is a Pettis integral of the pushforward of  onto _⊂ T_, as a direct consequence of Theorem <ref>.




Let x∈.
For U={x}, use Lemma <ref> with = and B_δ a ball of radius δ around x in T_, to get a sequence (y_δ^n)_n⊂ T_.
Then,

    lim sup_ncos∠(↑_^x,   ↑_^y_δ^n)=lim sup_n⟨ x,y_δ^n⟩_/d(,x)d(,y_δ^n)
       =1/d(,x)lim sup_n⟨ x,y_δ^n⟩_1/lim_nd(,y_δ^n)
       ≤1/d(,x)∫_B_δ^c⟨ x,y⟩_ d(x)/(B_δ)(B_δ)/(∫_B_δ∫_B_δ⟨ x,y⟩_ d⊗(x,y))^1/2.

Then, since ∫⟨ x,y⟩_ d(y)=0 by Lemma <ref>, letting δ→ 0, one gets

    1/(B_δ)∫_B_δ^c⟨ x,y⟩_ d(y)→ -d^2(,x).

and

    (∫_B_δ∫_B_δ⟨ x,y⟩_ d⊗(x,y))^1/2/(B_δ)→ d(,x)

Thus, 

    lim_δ→ 0^+lim sup_ncos∠(↑_^x,↑_^y_δ^n)=-1

One can thus choose (y̅^n)_n a sequence in (y_δ^n)_n,δ such that cos∠(↑_^x,↑_^y̅^n)→ -1.
Since T_ is a subspace of an Alexandrov space of curvature bounded below by 0, we also have 

    ∠(↑_^y̅^n,↑_^y̅^k)   ≤ 2π - ∠(↑_^y̅^n,↑_^x)-∠(↑_^x,↑_^y̅^k)
       → 0,

as n,k→∞.
Thus (y̅^n)_n correspond to a Cauchy sequence in the space of direction, and thus admits a limit in T_ - since its "norm" also admits a limit d(,x).
Finally, its limit y̅ satisfies cos∠(↑_^x,↑_^y̅)=-1, and therefore, it is the opposite y̅= -x.





Suppose (,d) is an Alexandrov space with curvature bounded below.
Then, for any probability measure ∈,

    ∫⟨ x,y⟩_ d⊗(x,y)≥ 0.

Moreover, if b^⋆ is an exponential barycenter of , then for all x∈,

    ∫⟨ x,y⟩_ d(y)=0.



For brevity, we will adopt the notation g for ∫ g d.

The result for  finitely supported is the Lang-Schroeder inequality (Proposition 3.2 in <cit.>).
Thus, we just need to approximate ⊗⟨ .,. ⟩_ by some _n⊗_n⟨ ,.,⟩_ for some finitely supported _n.

To do this, for i.i.d. random variable (X_i)_i of common law , denote _n the empirical measure.
Since  is not separable, we can not apply the fundamental theorem of statistics that ensures almost sure weak convergence of _n to .
However, for a measurable function f:×→, such that ⊗ f^2<∞, we get the following bound

    |⊗ f   -_n⊗_n f|^2
       = 1/n∑_ijkl [(⊗ f-f(X_i,X_j))(⊗ f-f(X_k,X_l)]
       =O(1/n),

since n(n-1)(n-2)(n-3) of the n^4 terms of the sum are equal to 

    [(⊗ f-f(X_1,X_2))(⊗ f-f(X_3,X_4))]=0.

And thus, _n⊗_n f→⊗ f in L^2(^⊗∞) and so there exists a (deterministic) probability measure _n supported on n points, such that _n⊗_nf→⊗ f.
We thus proved the first result applying f=⟨ .,.⟩_.

Now applying this first result to the measure _:=1/1+ + /1+δ_x, we get

    0   ≤ (1+)_⊗_⟨ .,.⟩_
       =⊗⟨ .,.⟩_+2⟨ x,.⟩_+^2x_^2.

Letting → 0^+, we get

    ⟨ x,.⟩_≥ 0.

Then equality follows from the hypothesis ⊗⟨ .,.⟩_=0 meaning that  is an exponential barycenter.






Let (,d) be an Alexandrov space with curvature bounded below.
Take ∈.
Let x_1,…,x_n∈ T'_ and U⊂ T_ bounded.
Then, for all >0, there exists y∈ T_ such that for all u∈ U,

    ⟨ y,u⟩_≤∑_i=1^n⟨ x_i,u⟩_ +,

and

    y^2≤∑_i,j=1^n⟨ x_i,x_j⟩_ +.






Let U⊂ T_ finite.
Take B⊂ measurable and a probability measure ∈ such that ⊗⟨ .,.⟩_=0 and (B)>0.
Then, there exists a sequence (y^n)_n such that for all u∈ U

    1/(B)∫_B^c⟨ u,x⟩_ d(x) ≥lim sup_n ⟨ u, y^n⟩_

and

    1/(B)^2∫_B∫_B⟨ x,y⟩_ d⊗(x,y)=lim_n d^2(,y^n).




Using the same arguments as in Lemma <ref>, we see that the empirical measures (_n)_n satisfy 

    _n⊗_n f→⊗ f,

in L^2(^⊗∞), for any f:×→.
In particular, taking f(x,y)=⟨ x,y⟩_1_B× B(x,y), the following convergence holds in L^2(^⊗∞),

    ∫_B∫_B⟨ .,.⟩_d_n⊗_n →∫_B∫_B⟨ .,.⟩_ d⊗,

and similarly for B^c.
Also, the law of large number ensures that almost surely, for all u∈ U,

    ∫_B⟨ .,u⟩_d_n →∫_B⟨ .,u⟩_ d,

and again, the same for B^c.
Thus, there exists a subsequence (of a deterministic realisation of) _n - that we rename _n - such that (<ref>) and (<ref>) both hold for all u∈ U.

Then, applying Lemma <ref> to finite sum

    1/(B)∫_B^c⟨ .,u⟩_d_n,

shows that there exists a sequence (y^n)_n∈ T'_ such that (<ref>) holds and for a sequence (_n)_n s.t. _n→ 0,

    y^n_^2≤1/(B)^2∫_B^c∫_B^c⟨ .,.⟩_d_n⊗_n+_n.

Then, applying the same Lemma <ref> again shows that there exists a sequence (z^n)_n⊂ T'_, such that

    0   ←1/(B)^2∫∫⟨ x,y⟩_ d_n⊗_n(x,y)
       =1/(B)^2(∫_B∫_B+∫_B^c∫_B^c+2∫_B∫_B^c)⟨ x,y⟩_ d_n⊗_n(x,y)
       ≥z^n_^2 + y^n_^2 + 2⟨ y^n,z^n⟩_-_n.

Letting n→∞, one obtains

    0   ≥lim_nz^n_^2+2⟨ y^n,z^n⟩_+y^n_^2
       ≥lim_n z^n_^2-2y^n_z^n_+y^n_^2
       =lim_n(z^n_-y^n_)^2≥ 0.

and which shows lim_n y^n=lim_nz^n and also that (<ref>) becomes an equality at the limit and therefore

    lim_nz^n_^2=1/(B)^2∫_B∫_B⟨ x,y⟩_d⊗(x,y)



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