Questions tagged [wasserstein]
The wasserstein tag has no usage guidance.
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There is a unique coupling between a probability distribution $\mu$ and a degenerate distribution $\mu_0$. [duplicate]
By degenerate distribution I intend https://en.wikipedia.org/wiki/Degenerate_distribution.
I cannot see why the set of couplings between some probability measure and a constant would be a singleton. ...
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1D-Wasserstein-Barycenter closed form
I just want to verify that my reasoning here is correct. It feels very basic but I can't seem to find this result in any textbook. It is well-known that the 1-Wasserstein distance in 1D can be ...
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Does this relation between Wasserstein distances hold: $W_1(\mu,\nu)\leq W_2(\mu,\nu) \leq ... \leq W_\infty(\mu,\nu)$?
I stumbled upon this interesting statement in this paper:
"One interesting observation is that the Wasserstein ambiguity
set with the Wasserstein order p = 2 is less conservative, because the 2-...
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Bound for expected value under Wasserstein metric
I'm reading a paper and the following result is presented:
$$ (\mathbb{E}_{F}[\|\mathbf{X}\|^k])^{1/k} \leq (\mathbb{E}_{F_{0}}[\|\mathbf{X}\|^k])^{1/k} + \epsilon, \ \forall F\in\mathcal{B}_{p}(F_{0},...
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Regularity of Kantorovich potentials for general cost function
I know De Philippis and Figalli have a paper studying the regularity of the Kantorovich potential. In Theorem 3.3, the authors show that the potential is $C^{k+2, \beta}$ if the density functions of ...
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proof that the wasserstein space is no manifold
This is my first question on this platform, I appreciate any suggestions on how to improve my question.
why is the Wasserstein space no manifold and in which way is its structure somehow similar to a ...
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Understanding different norms in the p-Wasserstein distance
The generalized p-Wasserstein distance, for $p\geq 1$, is given by
$$d_W(Q_1,Q_2):=inf \left\{\int_{\Xi_2}||\xi_1-\xi_2||^p \Pi(d\xi_1,d\xi_2)\right\}$$
where $\Pi$ is the joint distribution of $\xi_1$...
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Connect definition of 1st-order Wasserstein distance given CDFs and general definition of Wasserstein distance
In this post, the definition of the 1st-order Wasserstein distance is
$\int_{-\infty}^{\infty} |F_1 (x)-F_2(x)| dx$
In Wikipedia, I see something completely different.
How do I connect the 2 ...
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Showing that a minimzing sequence for the Wasserstein Variance functional must be tight
I am trying to understand this 2011 paper by Agueh & Carlier [https://www.ceremade.dauphine.fr/~carlier/AC_bary_Aug11_10.pdf] where they introduce the notion of barycenter in the 2-Wasserstein ...
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What is the image of a transference plan?
Let $\mu, \nu$ be two positive Borel measures on $\mathbb{R}^d$ with the same mass. A probability measure $\pi$ on $\mathbb{R}^d\times\mathbb{R}^d$ is called a transference plan from $\mu$ to $\nu$ if:...
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Quantitative bound on Wasserstein distances by $L^p$ distances?
Given two smooth probability densities $f$ and $g$ on $\mathbb{R}$ (or $\mathbb{R}_+$) with finite $p$-th moments. I am wondering if anyone is aware of some explicit upper bound on $W_p(f,g)$ in terms ...
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What is a wasserstein barycenter?
I am currently studying a paper on Wasserstein Fair Classification. Several places they mention the Wasserstein barycenter, weighted barycenter distribution or the Wasserstein barycenter distribution. ...
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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 \...
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Functional E convex and lower semicontinuous implies weakly lower semicontinuous in Wasserstein space
I have a certain functional $E : W_2 \rightarrow \mathbb{R}$, where $W_2$ is the 2-Wasserstein space (metric and separable).
Such functional is convex.
Now, can I state that if $E$ is strongly lower ...
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Sufficient Conditions on Metric Space for Wasserstein Distance?
For a metric space $M$ and the Wasserstein 1-distance $W_1$, what qualifying assumptions do we need for $M$? I have seen we only need $M$ be compact, $M$ be a Polish space, or $M$ be a Radon space. ...
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Wasserstein Metric Inequality
This is the exercise:
This exercise shows that “spreading out” probability measures makes them closer together. Define the convolution of a measure by: for any probability density function $\phi$, let ...
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Intuition of Wasserstein and Information geometry geodesics
Two important geometries that can be given to the space of multivariate Gaussian distributions are given by the Wasserstein distance and by the Fisher metric (ie. Information geometry).
Although there'...
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Are linear interpolation curves on Wasserstein spaces absolutely continuous?
Let $\mathcal{P}_2$ the space of absolutely continuous probability measures on $\mathbb{R}^d$ with finite second moment equipped with the $2$-Wasserstein metric.
Fix $\mu_0, \mu_1 \in \mathcal{P}_2.$
...
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how to solve for wasserstein duality easily in a special case when 2-Wasserstein inequality constraint is binding
I was going through this nice paper ” A Simple and General Duality Proof for
Wasserstein Distributionally Robust Optimization”, and one quick qu on applying Theorem 1 to my poject:
What if in my ...
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What is the conventional definition of $d(x,y)^{p}$?
In Optimal transportation, and more precisely in
"Optimal Transport: Old and New" (Definition. 6.1, on page 106 - actually on page 111 out of 998, in this link), the Wasserstein distance $...
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How to intuitively view Wasserstein distance dual as moving earth
The Wasserstein-1 distance can be viewed as the minimum amount of work needed to move one distribution to another distribution, as if the distributions were like piles of earth. The typical definition ...
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How to derive the $W_1$ Wasserstein distance written with the quantile functions (i.e. with inverse cumulative distribution functions) [closed]
INTRODUCTION. Either downloading the slides on Optimal Transport (OT) from the Marco Cuturi website (go to section "Teaching ENSAE --> OT --> Optimal Transport (Spring 2023)
[slides]" ...
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Optimal Mass Distribution Minimizing Average 2-Wasserstein Distance to a Set of Mass Distributions
Given a fixed set of $n$ points in 2D (Earth Movers distance Prpblem), $P = \{p_1, p_2, ..., p_n\}$, I am trying to find the mass distribution $\bar{M}$ that minimizes the average 2-Wasserstein ...
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Derive the $d=1$ form of the Wasserstein distance
QUESTION
Given
How to pass from the general definition of the Wasserstein distance (let's call it Equation (1)):
to the closed forms with d=1, here below (let's call them Equation (2) and Equation(3)...
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Is there a meaning of distances for 0<p<1 for Wasserstein distance?
The wikipedia link for Wasserstein metric is defined for $p\in[1,\infty)$.
https://en.wikipedia.org/wiki/Wasserstein_metric Given some data the distance can be calculated using an optimization ...
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Boundedness of $\dfrac{W_2(\mu_1+\varepsilon (\mu_2-\mu_1),\mu_1)}{\varepsilon}$ for 2 -Wasserstein metric
Let $\mathcal{P}_2(\mathbb{R}^{n})$ the space of Borel probability measures of finite second moment in $\mathbb{R}^{n}$ equipped with the $2$-Wasserstein metric $W_2$. Let $\mu_1$, $\mu_2 \in \mathcal{...
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Relations between Kolmogorov-Smirnov distance and Wasserstain distance
Let's have $X,Y \in \mathbb{R}$ with probability measures $\mu, \nu$, then the Kolmogorov-Smirnov distance is defined as follows
$$ d_K(X,Y)=\underset{x \in \mathbb{R}}{sup}\{|F_X(x) - F_Y(x)|\} $$
...
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Relation between Wasserstein distance and distribution convergence
Let's have a succession $X_n$ of real value random variable and another real value random variable X, then
$$ X_n \overset{d}{\xrightarrow{}} X \iff \lim_{n \to \infty}{}d_K(X_n,X) = 0 $$
where $d_K(X,...
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Proof of Lipschitz continuity of Wasserstein distance in WGAN paper
I'm reading the Wasserstein GAN paper(https://arxiv.org/abs/1701.07875)
by Martin Arjovsky et al.
My question is about the proof of the statement 2 of Theorem 1 in the paper.
Please see the appendix C....
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Is the space of probability measure dense in the space of probability measure with density?
Let $\mathcal{P}_2(\mathbb{R}^n)$ be the set of all probability measures with finite second moment. Let $\mathcal{P}_2^*(\mathbb{R}^n)$ be the subset of $\mathcal{P}_2(\mathbb{R}^n)$ such that any ...
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Relationship between optimal transport and gaussian kernel
Let's say P and Q be two different dirac delta probability measure, and suppose that $K_\sigma$ is a gaussian kernel. Let D be the wasserstein-2 distance.
It is known that $D(P,Q)=D(K_\sigma *P, K_\...
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Wasserstein distance between a distribution of a random variable and the distribution of its projection onto a subset of its sample space
Consider a random variable $x$ with a distribution $p_x$ supported on whole of $\mathbb{R}^n$ ($n$ being a natural number). Let $S \subset \mathbb{R}^n$. Let $y = {\rm proj}_S(x)$ denote the ...
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Does the reverse triangle inequality holds for Wasserstein-1 distance?
Let $(X,d)$ be a separable metric space with associated Borel $\sigma$-algebra $\mathcal{B}(X)$ and the set of Borel probability measures $\mathcal{P}(X)$. For $\mu,\mu'\in\mathcal{P}(X)$ Wasserstein-...
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Wasserstein distance inequality
Suppose $(\Omega, \mathcal F, \mathbb P)$ is a probability space. Suppose $X, X', Y, Y'$ are random variables.
Denote $W_1$ the Wasserstein-1 distance between $\mathbb P_X$ and $\mathbb P_{X'}$
and $...
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An inequality about the 2-Wasserstein distance
Let $W_2(\mu,\nu)$ denote the $2$-Wasserstein distance between two given probability measures $\mu$ and $\nu$ on $\mathbb R^n$. For a probability measure $\mu$ and $f:\mathbb R^n\to \mathbb R^n$, let $...
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Scaling property of the Wasserstein metric
I would need help with this example.
Let $(S, ||\cdot||)$ denote a normed vector space over $K =\mathbb R$ or $K =\mathbb C$. Let $X$ and $Y$ be $S$-valued random vectors with $E~[~||X||~] < \infty$...
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Advection reaction equation is solved by projection of solution of continuity equation
Suppose an absolutely continuous curve $\mu \colon (0, \infty) \to P_2(\Omega)$, where $P_2$ is the Wasserstein-2-space, fulfils the continuity equation
$$ \label{eq:CE} \tag{CE}
\partial_t \mu_t = \...
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Wasserstein-1 distance, $W(A\cdot B \| C\cdot D)=W(A \| C) + W(B \| D)$
For 4 independent random variables $A, B, C, D$ and Wasserstein-1 distance $W^1$,
$W^1(P_{A,B} \parallel P_{C,D})=W^1(P_A \parallel P_C)+W^1(P_B \parallel P_D)$
Does the above equation generally ...
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Optimal Transport between two Gaussians
Consider the optimal transport map $T$ between $N(\mu_0,\Sigma_0)$ and $N(\mu_1,\Sigma_1)$. I believed that the optimal transport was given by:
$$ T(x) = \mu_1 + \Sigma_1^{1/2} \Sigma_0^{-1/2}(x-\mu_0)...
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Why does it suffice to define the homogeneous projection operator by only testing with continuous instead of measurable functions?
In Lenaic Chizat's "Sparse Optimization on Measures with Overparametrized Gradient Descent" one finds the following definition: for a measure $\mu \in \mathcal P_2(\Omega)$ (the Wasserstein-...
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Can we bound the L1 distance between densities by Wasserstein distance of measures
Let $\mu_1$ and $\mu_2$ be two probability measures over a closed interval $[a, b]$, with respective density functions $\phi_1$ and $\phi_2$. Is there a way to bound the $L^1$ distance of the ...
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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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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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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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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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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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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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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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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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