Questions tagged [wasserstein-distance]

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

Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...
mnmn1993's user avatar
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Can we control the Wasserstein metric between $\mu$ and $\nu$ by their moment difference?

Fix $p \in [1, \infty)$. Let $(\mathcal P_p(\mathbb R^d), W_p)$ be the Wasserstein space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. Let $D_p$ be the collection of ...
Analyst's user avatar
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Some continuity issues of the optimal transport map (Brenier map)

Let $\mu$ and $\nu$ be two probability measures with finite moments (in $\mathcal{P}_2(\mathbb{R})$) equipped with 2-Wasserstein distance. Let $F_\mu$, $F_\nu$ be their cumulative distribution ...
mnmn1993's user avatar
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Gradient flows: evolution of geodesics

I’m trying to understand if, when I move the marginals of a Wasserstein geodesic along a contractive flow, the geodesic between the new probability measures is “near” to the geodesic connecting the ...
Ciccisio's user avatar
4 votes
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226 views

Perturbation of Wasserstein distance: looking for references

I am doing readings related to Optimal transport which is new to me and I often encounter the following statement regarding a sort of derivative of the Wasserstein distance: $u$ and $v$ be two ...
Guy Fsone's user avatar
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Construct a Bregman divergence from Wasserstein distance

I was wondering whether one has studied the Bregman divergence arising from a squared Wasserstein distance. More precisely, let $\Omega\subset \mathbb{R}^d$ be a compact set and $c\in \Omega\times \...
John's user avatar
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2 answers
172 views

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

For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let $$ \...
Akira's user avatar
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1 answer
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Lipschitz-type inequalities for Markov kernels

Let $K(\cdot\mid\cdot)$ be a Markov kernel. For a measure probability $\mu$, denote by $\mu K$ the probability measure induced by $K$ and $\mu$, i.e. $(\mu K)(A):=\int_\Omega \mu(d\omega)K(A\mid\omega)...
Michele's user avatar
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3 votes
2 answers
563 views

Wasserstein distance between product measures

For two probability measures $\mu,\nu$ on $\mathbb{R}^n$, let $$W_p(\mu,\nu)^p=\inf_{\pi}\int\|x-y\|^p_2\ \pi(dx\ dy)$$ denote the $p^\text{th}$ Wasserstein distance between $\mu, \nu$, where the ...
Ribhu's user avatar
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Optimal transport between two matrices

I'm investigating the use of optimal transport to define a distance between non-negative matrices that satisfy the condition $\mathbf e^\top\mathbf M\mathbf e = 1$ (i.e., the sum of all elements in ...
Peyman's user avatar
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Derivative of Wasserstein distance $W^p_p$ along solutions of the continuity equation (contradicting statements in different sources)

Let $(\rho^{(i)}_t,{\bf v}^{(i)}_t)$ for $i = 1,2$ be two solutions of the continuity equation $$\partial \rho^{(i)}_t + \nabla\cdot \left({\bf v}^{(i)}_t \rho^{(i)}_t\right) = 0 \label{1}\tag{1}$$ on ...
Fei Cao's user avatar
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0 answers
161 views

Implicit function theorem in Riemannian manifold and Wasserstein space

My question is about to what extent can we extend the implicit function theorem to Riemannian manifolds. In the Euclidean space, consider a bivariate function $F \colon \Theta \times \mathcal{X} \...
Steve's user avatar
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2 answers
183 views

Getting Wasserstein closeness from a derivative estimate

In my setting, $\mu$ and $\nu$ are probability measures on $\mathbb{R}^{2}$ with compact support. For any function $f\in{C^{2}_{b}(\mathbb{R}^{2})}$, I have the estimate: $$ |\mathbb{E}_{\mu}(f)-\...
David Pechersky's user avatar
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1 answer
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Is the Wasserstein distance to the empirical measure minimized by the underlying distribution?

Let $S$ be a metric space and denote the set of probability measures on $S$ by $\mathcal{P}(S)$. Fix $\mu\in \mathcal{P}(S)$ and denote the law of $N\geq 1$ i.i.d samples $X=(X_1,\ldots,X_N)$ from $\...
joemrt's user avatar
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Concentration inequalities for random measures

For random variables $X_1,\dots,X_n$ with common mean $\mathbb{E}[X_i]=\mu$ and common bounds $a\leq X_i\leq b$, we have the very useful Hoeffding's inequality: $$\mathbb{P}\left(\left|\mu -\frac1n\...
Tyler6's user avatar
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1 answer
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An inequality involving the Wasserstein distance and chi-squared distance

$\newcommand{\N}{\mathbb N}$Let $P$ be the set of all probability mass functions on $\N_0:=\{0\}\cup\N$, where $\N:=\{1,2,\dots\}$. Let $P_{>0}$ denote the set of all $q=(q_0,q_1,\dots)\in P$ such ...
Iosif Pinelis's user avatar
2 votes
1 answer
186 views

Upper bound Wasserstein distance by $\chi^2$ distance

Given two random variables $X,Y$ which are both $\mathbb{N}$-valued and have the same expected value (which is some fixed positive constant), and denote their probability mass functions by ${\bf p} = (...
Fei Cao's user avatar
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How to find the point at minimal average distance of a given measure

Given a compactly supported probability measure $m$ on $\mathbb{R}^n$, we can define its average distance to a point $x$ as $\int_\mathbb{R^n}d(x,y)dm(y)$. In this question I found that for a given ...
Saúl RM's user avatar
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Rates of convergence of empirical measures in Wasserstein distance

Let $X_1, X_2, \ldots$ be iid random variables in $\mathbb {R}^d$ with common distribution $\mu$, and $\mu_N = \frac 1N \sum_{k=1}^N \delta_{X_k}$, $N \geq 1$, the associated empirical measures. If $\...
Leonid Beya's user avatar
3 votes
1 answer
130 views

Equivalent definition of the Kantorovich-Fisher-Rao distance

I am reading this paper "A JKO splitting scheme for Kantorovich-Fisher-Rao gradient flows" (https://arxiv.org/abs/1602.04457) and in the proof of Proposition 2.2, basically, if the measure ...
Sean's user avatar
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1 answer
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Distance between empirical measures and thickened version

Let $\mathcal{H}$ be a separable Hilbert space and let $x_1,...,x_n$ be points in $\mathcal{H}$. Let $\varepsilon >0 $ be given and consider the measures $$ \mu := \frac1{n}\,\sum_{i=1}^n\, \...
ABIM's user avatar
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1 vote
1 answer
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Bounding $2$-Wasserstein distance and the $L^1$ distance

My questions come from the paper Logarithmic Sobolev inequalities for some nonlinear PDE’s written by F. Malrieu (May 2001) where author omitted a good amount of details to be filled. Suppose that $W$ ...
Fei Cao's user avatar
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4 votes
1 answer
252 views

2-Wasserstein metric on convolution of probability distributions

I have two related questions. Let $\mu$ and $\nu$ be two distinct probability measures on $\mathbb{R}^n$ with finite second moments, and $W_2(\cdot,\cdot)$ be the $2$-Wasserstein metric. The question ...
F J's user avatar
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4 votes
1 answer
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Gradient of Wasserstein distance in the sense of Otto's calculus

I am learning the idea of "gradient" of a functional in Otto's calculus. It is defined as follows. Suppose the space we are thinking about is $(\mathcal{P}_{2,AC}(\mathbb{R}^d),W_2)$, the ...
MikeG's user avatar
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2 votes
1 answer
727 views

Convergence of empirical measures in Wasserstein distance

Let $X_1, X_2, \ldots$ be iid random variables with common distribution $\gamma$, the standard Gaussian distribution on $\mathbb {R}$, and let $\mu_n = \frac 1n \sum_{i=1}^n \delta_{X_i}$, $n \geq 1$, ...
Josh Ibrahim's user avatar
6 votes
0 answers
863 views

Wasserstein distance and Monge-Kantorovich-Rubinstein duality

The definition of Wasserstein $p$-distance between two measures $\mu$ and $\nu$ 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
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1 vote
1 answer
309 views

Are there any results on concentration bounds of Wasserstein distances between empirical measures?

I know there are concentration bounds on $W(\mu,\hat{\mu})$ where $\mu$ and $\hat{\mu}$ are true and empirical distributions respectively, but is there anything out there on $W(\mu,\nu)$ versus $W(\...
Kashif's user avatar
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2 votes
1 answer
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Closed-form formula for Wasserstein distance between uniform discrete distribution and discrete distribution with same support

Let $x_1,\ldots,x_n$ be $n \ge 1$ distinct points in $\mathbb R^d$ and consider two discrete distributions on these points $\mu = (1/n)\sum_{i=1}^n\delta_{x_i}$, and $\nu = \sum_{i=1}^n\nu_i\delta_{...
dohmatob's user avatar
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5 votes
2 answers
7k views

Upper bound total variation by Wasserstein distance for continuous distance

I am reading the survey of the relationships between metrics of distributions (see https://arxiv.org/pdf/math/0209021.pdf for the paper). The general results show that for general distributions, we ...
Felix Y.'s user avatar
  • 123
23 votes
6 answers
10k views

Metrization of weak convergence of signed measures

Edit: Changed from "Hausdorff" to "metric" spaces. Let $\mathcal{M}(\Omega)$ denote the space of signed regular Borel measures on a compact metric space $\Omega$. By Riesz-Markov, ...
Dirk's user avatar
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11 votes
2 answers
2k views

Wasserstein distance in R^d from one dimensional marginals

This question occurred to me while I was reading Klartag's papers on central limit theorems for convex bodies. Given probability measures $\mu$, $\nu$ on (the Borel $\sigma$-field of) $R^d$ with ...
Roberto Imbuzeiro Oliveira's user avatar