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(Sub)Optimality of random transport

Problem Setup: Consider the intervals IR=[aR,bR] and IB=[aB,bB]. Let FR and FB denote the CDF of distributions with support on the intervals IR and IB. I draw k red and ...
yfful's user avatar
  • 25
2 votes
1 answer
129 views

Wasserstein distance of push-forward measures

I asked this same question on MSE, but with no luck, so I am trying to ask here. Consider two measures μ,ν on Rn. Now consider a map (a priori only measurable, but feel free to add ...
tommy1996q's user avatar
0 votes
0 answers
52 views

Path-homotopy in Wasserstein space

Consider two vector fields b0,b1C2([0,1]×Rd;Rd) and the solutions ρ0,ρ1AC([0,1];P2(Rd)) to the associated Fokker-Planck equations $$...
Ciccisio's user avatar
0 votes
0 answers
39 views

Comonotone solution for Optimal Transport problems with supermodular surplus

In Alfred Galichon's book Optimal Transport Methods in Economics the foollowing result is stated for OT problems on the real line. Theorem 4.3.(i) Assume that Φ is supermodular. Then the primal ...
Francesco Bilotta's user avatar
0 votes
1 answer
104 views

Sequential compactness of a sequence of curves of Borel probability measures

$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \DeclareMathOperator*{\supp}{supp} \newcommand{\bR}{\mathbb{R}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bP}{\mathbb{...
Akira's user avatar
  • 825
1 vote
1 answer
118 views

Is there ε(0,1) such that supt[0,ε][t]β<?

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bD}{\mathbb{...
Akira's user avatar
  • 825
0 votes
0 answers
55 views

Any useful bases for the topology induced by the t-Wasserstein distance?

I am working on Rd equipped with the usual Euclidean metric. I know of one nice base for Wt, namely: $$\left\{ B_p (r) : r>0, p=\sum_{i=1} ^n \alpha_i \delta_{x_i},\text{ ...
J.R.'s user avatar
  • 291
1 vote
1 answer
83 views

Lipschitz approximation of a probability measure with finite 1-st moment by the ones with finite p-th moment

For p[1,), let Pp(Rd) be the space of Borel probability measures on Rd with finite p-th moment. We endow Pp(Rd) with the ...
Akira's user avatar
  • 825
0 votes
0 answers
114 views

Some stability and estimate of the optimal transport map (Brenier map)

Let μ and ν be two probability measures with finite moments (in P2(R)) equipped with 2-Wasserstein distance. Let Fμ, Fν be their cumulative distribution ...
mnmn1993's user avatar
2 votes
0 answers
95 views

Can we control the Wasserstein metric between μ and ν by their moment difference?

Fix p[1,). Let (Pp(Rd),Wp) be the Wasserstein space of all Borel probability measures on Rd with finite p-th moment. Let Dp be the collection of ...
Analyst's user avatar
  • 657
0 votes
1 answer
211 views

Some continuity issues of the optimal transport map (Brenier map)

Let μ and ν be two probability measures with finite moments (in P2(R)) equipped with 2-Wasserstein distance. Let Fμ, Fν be their cumulative distribution ...
mnmn1993's user avatar
0 votes
1 answer
88 views

Can we lower bound this entropy by Rdρk(x)dx and Rd|x|2ρ(x)dx?

We define U:[0,)[0,) by U(0):=1 and U(s):=slogs+(1s) for s>0. Then U is strictly convex. The minimum of U is 0 and is attained at s=1. Let $\mathcal ...
Akira's user avatar
  • 825
4 votes
2 answers
256 views

Are the sublevel sets of Boltzmann entropy compact in Wasserstein distance?

For Lebesgue-absolutely continuous probability measures ρLd in the whole space Rd with finite second moments (i-e ρP2ac(Rd)), let $$ \...
Akira's user avatar
  • 825
4 votes
0 answers
197 views

Compactness of the unit ball in the space of Radon measures w.r.t. the Kantorovich-Rubinstein norm

This question was posted previously but has not attracted any responses so I am repharising it in a slightly different language hoping to reach a wider community Let (X,d) be a pointed metric space ...
Yury Korolev's user avatar
1 vote
0 answers
169 views

Optimal transport-like problem where the objective depends on conditional probability distribution

I would like to know if the following problem can be studied as an optimal transport problem, possibly imposing additional assumptions on the data. Consider two sets $\...
Francesco Bilotta's user avatar
2 votes
0 answers
49 views

σ-compactness of probability measures under a refined topology

Denote Polish spaces (X,τx) and (Y,τy), where X and Y are closed subsets of R. Consider a Borel measurable function $f: (X \times Y, \tau_x \times \tau_y) \rightarrow \...
Hans's user avatar
  • 195
4 votes
1 answer
668 views

Optimal Transport: how is this transport map Borel measurable?

I'm reading Theorem 1.17. and its proof at page 14 of Santambrogio's Optimal transport for applied mathematicians. The content is not hard but a little bit long (because of related detail). Please ...
Akira's user avatar
  • 825
1 vote
1 answer
243 views

Continuity equation for a density of a measure

From the paper of Ambrosio-Crippa, it is known that if β:Rd×[0,T[Rd is suitably regular, then the system $$ \begin{cases} \dfrac{\partial\mu}{\partial t}(x,...
Redeldio's user avatar
  • 171
3 votes
0 answers
278 views

Radon-Nikodym derivative of vector-valued measure with respect to another vector-valued measure

Let (X,||) be a Banach space. I am interested in whether one can extend the definition of the Kullback-Leibler divergence $$ \text{KL}(\mu \ \Vert \ \nu) := \int_{\Omega} \ln\left(\frac{\...
ViktorStein's user avatar
4 votes
1 answer
360 views

Measurability of Markov kernel wrt the Borel σ-algebra generated by the weak topology

Consider two Polish metric probability spaces (A,ΣA) and (B,ΣB), endowed with their Borel σ-algebras. Denote as $\mathcal{P}_\mathcal{B}...
ECL's user avatar
  • 345
2 votes
0 answers
82 views

Estimate of Wasserstein distance and flow of vector fields under particular assumptions

Let μ be a compactly supported absolutely continuous probability measure. Let v,u be Lipschitz vector fields. For a vector field w recall that Φwt denotes its flow. A classical estimate ...
Jun's user avatar
  • 303
0 votes
2 answers
201 views

Couplings as generalized functions

I've been casually reading about optimal transport, and I was intrigued by the Wasserstein metric, in which we define the distance between two measures μ and ν on a metric space X by $$ W_p(\...
Danny Stoll's user avatar
3 votes
2 answers
295 views

Wasserstein convergence of "series expansion'' of probability measure

Let X be a Polish space and let (μi)i=1 be a sequence of probability measures in the Wasserstein space P(X) on X. Let (βi)i=1 be a summable ...
ABIM's user avatar
  • 5,405
0 votes
1 answer
275 views

Upper bound on the p-Wasserstein distance Wp(ξ,aξ) for some constant a0

Let ξ be a random vector taking values in Rd. Is there an upper bound on the p-Wasserstein distance Wp(ξ,aξ) for some constant a0? I have seen that if $p=...
Juan Miguel Morales González's user avatar
1 vote
0 answers
130 views

Orthogonality in Wasserstein tangent space for discrete measures with equal mass

Let say I have N discrete probability measures (μ1,...,μN) where each of them has n points in R2 of equal mass. Let P(X) be the space of these probability measures ...
Jean Hugue's user avatar
2 votes
1 answer
173 views

Is there an analogue of transportation-cost inequality under a weighted Log-Sobolev Inequality?

It is known that under the Log-Sobolev Inequality for π, i.e., if for all ρ, $$H_\pi(\rho):=\int \rho(x)\log\frac{\rho(x)}{\pi(x)}dx \leq \frac{1}{2\beta}\int \rho(x)\left\|\nabla \log\frac{\...
user_qj's user avatar
  • 21
4 votes
1 answer
396 views

Is the optimal transport of radially symmetric measures also radially symmetric?

Let μ and ν be radially symmetric probability measures on Rd. Consider the Kantorovich optimal transport problem between μ and ν, with convex, nonnegative cost. Suppose ...
Nate River's user avatar
  • 6,323
0 votes
1 answer
86 views

Is integration against an indicator Wasserstein-Continuous

Let Pp(X) denote the Wasserstein space over a compact metric space X, and 1p<. Fix a non-empty closed subset CX. Then is the map: $$ \mathbb{P} \mapsto \...
ABIM's user avatar
  • 5,405
2 votes
0 answers
302 views

Simplify Kantorovich–Rubinstein duality when distributions share a common marginal

Consider the product of two metric spaces X×Y, and two probability distributions μ and ν on this product space. By the Kantorovich-Rubinstein duality, I can write the Wasserstein-1-...
joemrt's user avatar
  • 53
8 votes
0 answers
1k views

Wasserstein distance and Monge-Kantorovich-Rubinstein duality

The definition of Wasserstein p-distance between two measures μ and ν on a Polish space X is given by $$ W_p(\mu, \nu)^p = \inf_{\gamma \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^p\; d\...
SMS's user avatar
  • 1,407
1 vote
1 answer
241 views

Continuity of pushforward operation

Let X and Y be compact metric spaces and let f,g:XY be ϵ-uniformly close; i.e.: supxXdY(f(x),g(x))<ϵ. Then, are their push-forwards close in ...
ABIM's user avatar
  • 5,405
0 votes
0 answers
87 views

How does one define weak convergence of probability measures in L(Ω)?

I am reading the following article and on page 9/17 (above Eqn (4.9)) the authors state that if γϵk|_Gδ×Ωγ|_Gδ×Ω as ϵk0 ...
Student's user avatar
  • 557
1 vote
1 answer
1k views

Closed-form upper-bounds for Wasserstein distance between finite measures

Let x1,,xn,y1,,ynR and such that xixj and yiyj if ij. Let a,b be elements of the probability n-simplex. Define the measures $\mu\triangleq \...
ABIM's user avatar
  • 5,405
1 vote
1 answer
247 views

Scaling behavior of Wasserstein distances

Let p>1 and μν be two probability measures on ΩRd a bounded set. For α0, we let $$C_\alpha(\mu,\nu) = \inf_\sigma \frac{W_p(\mu+\sigma,\nu+\sigma)}...
Vincent's user avatar
  • 83
2 votes
0 answers
146 views

Prove the equicontinuity of a maximizing sequence

Let X be a compact subset of R and c(x1,x2,x3,x4) be a fixed bounded continuous functions on X4. Assume μ,ν are probability measures on X2, and μν is the ...
aurora_borealis's user avatar
1 vote
1 answer
719 views

Transport of measure

Let's disintegrate μ and ν, two probabilities on Rd , according to πk(x1,...,xd)=(xk,...,xd) We get a family of measures and each measure $\mu_{k,d}^{+...
CechMS's user avatar
  • 179
0 votes
0 answers
75 views

Optimizer of a semi-discrete optimal transport problem

Provided two probability distributions μ(dx)=ρ(x)dx and ν(dx)=ni=1piδyi(dx) that are supported on some measurable set ΩRd, we consider the semi-...
user avatar
6 votes
1 answer
580 views

Why is it difficult to solve the Monge problem directly?

I'm trying to understand something about the Monge problem. The Monge problem is: Let c(x,y):Rd×RdRd and $$\mathcal{T}(\mu_1,\mu_2) = \{ T: \...
yoshi's user avatar
  • 427
1 vote
0 answers
60 views

Euler-Lagrange Equation for Kantorovich Dual Problem

Given two probability measures μ and ν, the Kantorovich Dual problem for quadratic cost is to minimizeϕ(x)dμ+ψ(y)dν over pairs $(\phi,\psi)\in L^1(d\...
Ryan's user avatar
  • 325
1 vote
0 answers
97 views

Dependency of the Wasserstein metric on its parameters

Let the population on some region ΩRd be modeled by a density function ρ:Ω(0,+). Provided n1 food trucks labeled by their capacity $p_1,\ldots, p_n\in (...
user avatar
4 votes
0 answers
187 views

Transport Distance between Level Sets of a Convex Function

Suppose I have a well-behaved, strictly convex function f:Rd[0,), and assume that f has its unique minimiser at x=0, with f(0)=0. For y>0, I define the ...
πr8's user avatar
  • 801
2 votes
1 answer
317 views

optimal transport, measurable selection

Let X=Y=Rd and let ν be a probability measure on Rd. Consider the collection of probability measure π on X×Y such that π has y-marginal ν: $$ \Pi(...
Ryan's user avatar
  • 325
4 votes
2 answers
415 views

Effect of perturbing the atoms of a measure on the Wasserstein distance

Let (X,d) be a metric space, x1,,xNX and x1,,xNX be atoms, and G=Ni=1piδxi, G=Ni=1piδxi, and $G''=\sum_{i=1}^Np_i'\delta_{x_i'}...
JohnA's user avatar
  • 710
2 votes
0 answers
116 views

A variant of the optimal transport

Let μ, ν and γ be three probability measures on R. Consider the optimisation problem as follows: inf(X,Y,Z) E[|YZ|2], where the inf is taken ...
user111097's user avatar
3 votes
0 answers
106 views

Dependency of the Wasserstein distance on the parameter: a differential perspective

Let μ(dx)=ni=1piδxi(dx) and ν(dy)=ρ(y)dy be two probability measures on Rd. Consider the 2Wasserstein distance below: $$W_2(\mu,\nu)^2 \quad := \quad \inf_{\...
user111097's user avatar
3 votes
0 answers
243 views

Parametric distances on product spaces of measures

Disclaimer: Please excuse my loose language. I'm neither an expert in geometry nor probability. Please ask for clarification if something appears unclear or awkward to you. Let X be a topological ...
dohmatob's user avatar
  • 6,853
4 votes
0 answers
757 views

Tangent space and gradient on subspace of Wasserstein space given by finitely supported measures

Let P2(M) be the 2-Wasserstein space over some Riemannian manifold (M,g) (connected, complete, and without boundary). Let FP2(M,n) be the subspace of probability measures ...
S.Surace's user avatar
  • 1,675
2 votes
1 answer
309 views

Density in Wasserstein space

I am wondering whether the following result is true: Let Wp(Rd) be the Wasserstein space of order p and let η and γ be two probability measures in $\mathcal W_p(\...
Ryan's user avatar
  • 325
8 votes
3 answers
937 views

Question about Wasserstein metric

Let μ and ν be two probability measures on Rn with finite first moment. Denote by d:=W1(μ,ν), where W1(,) stands for the Wasserstein distance of order 1. My ...
user111097's user avatar
8 votes
1 answer
623 views

Completion of spaces of measures w.r.t. weak norms

For the sake of concreteness denote by M0(X) the linear space of all signed Borel measures σ with σ(X)=0 on some metric space (X,d) and fix some base point x0X. On this space ...
Dirk's user avatar
  • 12.7k