Questions tagged [wasserstein]

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Absolutely continuous curves in Wasserstein distance and measurability.

Let $(X, d, \mu)$ be a metric measure space. Let $P^1(X)$ denote the space of probability measures on $(X,d)$, which have finite first moments, that is: \begin{equation} \nu \in P^1(X) \implies \int d(...
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1 answer
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Weighted median of distribution functions

I am working on the following barycenter problem: Suppose we are given $N>1$ probability measures on $\mathbb{R}$ with cumulative distribution functions $F_1,\dots,F_N$ and weights $a_1, \dots, a_N ...
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1 vote
1 answer
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Is the median of CDFs again a CDF?

I am working on the following barycenter problem: Suppose we are given $N$ probability measures on $\mathbb{R}$ with cumulative distribution functions $F_1,\dots,F_N$ and we are interested in the ...
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1 vote
0 answers
80 views

Specific Wasserstein distance on $\Bbb{R}^2$

Consider the $W^2$ Wasserstein distance on $\Bbb{R}^2$, which we take with its Euclidean metric. Given a probability measure $p$ on $\Bbb{R}$, consider the following two couplings of $p$ with itself: ...
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4 votes
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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\...
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40 views

Wasserstein distance of convolution of measures

Let $\mu_1, \mu_2,\nu_1,\nu_2$ be measures on $\mathbb{R}^d$. It is well known that the $p$-Wasserstein distance satisfies $$\mathcal{W}_p(\nu_1 *\mu_1, \nu_1 *\mu_2) = \mathcal{W}_p(\mu_1,\mu_2),$$ ...
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17 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 ...
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67 views

Derivative of the Wasserstein Metric between two Gaussians

I am trying to take the derivative of the squared Wasserstein metric between two Gaussian probability densities, which is given by $W_2^2(q_0, q_1) = \| \boldsymbol{\mu}_0 - \boldsymbol{\mu}_1 \|_2^2 +...
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2 votes
1 answer
51 views

Prove $\frac{1}{n}∑_{k=1}^n \Big(x_k^2 + 2\sqrt{3}(1 - \tfrac{2k-1}{n}) x_k + 1 \Big)>0$ given $x_1<…<x_n$

I know for a fact that $$\frac{1}{n}∑_{k=1}^n \Big(x_k^2 + 2\sqrt{3}(1 - \tfrac{2k-1}{n}) x_k + 1 \Big)>0 \qquad\text{if $x_1<x_2<…<x_n$}$$ should hold because I derived this sum as the ...
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Measuring the similarity between distance vectors

I am trying to measure the correlation between a probability distribution and a scalar value. For instance, I have the following: Vector of values Corresponding Scalar Vec 1 Scalar 1 Vec 2 Scalar 2 ...
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23 views

How about $p$ is a rational number or irrational in definition of the Wasserstein distance?

Let $\mu$ and $\nu$ be two Borel probability measures on $R^d$ and let $1\le p<\infty$. The Wasserstein distance of order $p$ between $\mu$ and $\nu$ is defined by \begin{equation} W_p(\mu,\nu):=(\...
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1 vote
1 answer
52 views

What is the p-th moment finite in the definition of Wasserstein space?

I am confused about the following notation: For a simple case, let $X=R^d$ or $X=R$. What dose $$ \int_X \|x\|^pd\mu(x) $$ mean for a Borel probability measure $\mu$? For $X=R$, then $x\in R$ is a 1-...
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Equivalence of Earth Mover's Distance (Wasserstein) and $L_1$ norm

I'm curious if the Earth Mover's Distance (EMD) metric is perfectly embeddable with no distortion in $L_1$ space. A paper from 2003 suggests there will be distortion, but I have heard that EMD and $...
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Is there any relationship between the following two expectations?

Is there any relationship between the following two expectations? $\mathbb{E}_{\mathbb{Q}}[\|\boldsymbol{\tilde{\xi}} - \boldsymbol{\tilde{\xi}}^{\prime}\|]$, and $\mathbb{E}_{\mathbb{Q}}[\|\mathbf{A}...
1 vote
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30 views

Derivative of Kantorovic Potential wrt to Measure

The Kantorovich Dual of the 1-Wasserstein distance $W_1(p,q)$ between two densities $p(x), q(x)$ is given by $$W_1(p,q) = \sup_{|f|_L\leq 1} \int f(x)(p(x)-q(x))dx$$ with $|f|_L$ denoting the ...
0 votes
1 answer
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Question on Kantorovich-Rubinstein Duality proof

I am currently working on understanding the Kantorovich-Rubinstein duality and Wassertein loss. The following part of these class notes: Collecting the terms algebraically we can rewrite the ...
1 vote
1 answer
166 views

Wasserstein metric vs Holder continuity

It is well known that if $f$ is a Lipschitz continuous function, i.e. $$\forall x,y\in \Omega\qquad |f(x)-f(y)|\le L\|x-y\|$$ then, for any two probability distributions $\mu, \nu$ $$\int_\Omega f(x)(...
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Calculating the limit of a Wasserstein distance of two SDE's

I am trying to prove that: $\lim_{t \to \infty} W_2(\mu_t, \nu_t) = 0 $ where we have that $\mu_t = Law(X_t)$ and $\nu_t = Law(Z_t)$ with $$dX_t = -h(X_t)dt + \sqrt(\frac{2}{\beta})dB_t$$ $$dZ_t = -h(...
1 vote
0 answers
73 views

2-Wasserstein barycenter of uniform distribution on ellipsoid

Let $A$ be a positive-definite symmetric matrix. Consider the ellipsoid $E = \{ x \in \mathbb{R}^n \colon <x A^{-1} x> \leq 1 \}$. Now consider uniform distribution $\mu_1, \ldots, \mu_n$ on $...
0 votes
1 answer
56 views

What is a clever or efficient way to compute this variant of the Wasserstein distance between persistence diagrams?

A two-dimensional persistence diagram in $[0,1]$ say is just a multiset of points of $\mathbb R^2$. Given two diagrams $P=\{p_1=(a_1,b_1),\ldots, p_n= (a_n,b_n)\}$ and $Q=\{q_1=(c_1,d_1),\ldots, q_n= (...
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1 vote
1 answer
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Extension of Kantorovich-Rubinstein inequality.

Let $(\mathcal{X}, \Sigma)$ be a Polish metric space, endowed with the Borel $\sigma$-algebra. Let $\mathscr P$ be the space of probability measures on $\mathcal X$ and $\mathscr P^1$ be defined as $$\...
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4 votes
1 answer
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Maximiser of $W_1(\mu, \nu)$ can be changed outside of $\text{conv}(\text{supp}(\mu) \cap \text{supp}(\nu))$ (under additional assumptions)

Let $(X, \| \cdot \|)$ be a reflexive Banach space and $\mathbb{P}_n$, $\mathbb{P}_r$ be measures on $X$. Let the support of $\mathbb{P}_r$, $M := \text{supp}(\mathbb{P}_r)$ be a weakly compact set ...
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1 vote
0 answers
57 views

Density associated with Wasserstein geodesic

Suppose we have two absolutely continuous distribution functions $F$ and $G$ with densities $f$ and $g,$ respectively. Assuming a quadratic cost function, the Wasserstein geodesic at time $t$ is the ...
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3 votes
1 answer
102 views

Will $L^1\log L^1$ bound gives strong $L^1$ convergence?

I am trying to prove this statement: given a sequence $\{f_n | f_n > 0, c_1 \leq \int_{\Omega} f_n \log{f_n} \leq c_2 \}$, here $\Omega$ is a bounded domain; can we prove the $L^1$ strong ...
3 votes
1 answer
62 views

Covering with sets of negligible boundary

I am studying causality theory in Lorentzian length spaces, and I have a question about geometric measure theory in general (it will help me with a proof I am trying to finish): Suppose we have a ...
1 vote
0 answers
93 views

Change of variables - Wasserstein distance

My question is how exactly the change of variable formula is applied in the second row of the equation?
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1 answer
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Computing the Wasserstein $1$-Distance of the distribution $P_{0} \sim (0, Z)$ for $Z \sim U[0,1]$ and $P_{\theta} := (\theta, Z)$.

In reading about Wasserstein GANS here, an example is given in the middle of page $4$. Let $Z \sim U [0,1]$, and let $P_0 \sim [0,Z] \in \mathbb{R}^2$. Let $P_{\theta} := (\theta, Z)$ be a probability ...