Questions tagged [optimal-transportation]
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260 questions
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How to control Wasserstein distance in terms of characteristic function
Let $\mathcal P(\Omega)$ be the set of probability measures supported on some compact subset $\Omega\subset\mathbb R^d$. For $\mu\in\mathcal P(\Omega)$, denote by $F_{\mu}$ its characteristic function,...
user128095
0
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2
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How to compute the unique disintegration w.r.t. the first coordinate?
Set $\pi=\frac{1}{4}(\delta_{(1,0)}++\delta_{(1,3)}+\delta_{(1,1)}+\delta_{(2,2)})$. Suppose that $\pi\in\Pi(\mu,\nu)$.
How to get the disintegration of $\pi$ with respect to $\mu$?
- 288
3
votes
2
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295
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Wasserstein convergence of "series expansion'' of probability measure
Let $X$ be a Polish space and let $(\mu_i)_{i=1}^{\infty}$ be a sequence of probability measures in the Wasserstein space $\mathcal{P}(X)$ on $X$. Let $(\beta_i)_{i=1}^{\infty}$ be a summable ...
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2
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0
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Optimal transport: how is the use of disintegration theorem valid in this construction of $\widetilde{\phi}$?
Let $X,Y$ be Polish spaces and $\mathcal P(X)$ the space of all Borel probability measures on $X$.
Fix $\mu\in \mathcal P(X), \nu \in \mathcal P(Y)$. Let $\pi \in \Pi(\mu, \nu)$, i.e., $\pi \in \...
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2
votes
1
answer
304
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An approximation problem w.r.t marginal distribution of coordinates of uniform random vector on high-dimensional unit-sphere
Let $X=(X_1,\ldots,X_d)$ be uniformly distributed on the sphere of radius $\sqrt{d}$ in $\mathbb R^d$. Fix a "sufficiently integrable" function $h:\mathbb R \to \mathbb R$, and define ...
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1
vote
1
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Does the topology of Wasserstein space $(\mathcal P_p (E), W_p)$ coincide with the initial topology induced by $\mathcal C_b(E) \cup \{g_p\}$?
Let $(E, d)$ be a Polish space and $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$. Let $p \in [1, \infty)$ and $\mathcal P_p (E)$ the space of all Borel probability ...
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1
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1
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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\, \...
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3
votes
1
answer
385
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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(\...
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2
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Building the Wasserstein space by pushforwards
Let $\mathbb{R}^d$ denote the $d$-dimensional Euclidean space, $\mathcal{W}_2(\mathbb{R}^d)$ denote the $2$-Wasserstein space with respect to the $d$-dimensional Euclidean space $\mathbb{R}^d$. Let $...
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5
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2
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Comparison of Information and Wasserstein Topologies
There are many possible metrics one can place on the space of Gaussian probability measures on $\mathbb{R}^n$, with strictly positive definite co-variance matrices. Let's denote this space by $X$.
I'...
1
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1
answer
306
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When are Wasserstein spaces $CAT(\kappa)$?
Let $(X,d)$ be a complete and separable metric space and, for $1\leq p<\infty$, let $(\mathcal{P}_p(X,d),W_p)$ be the $p$-Wasserstein space on $(X,d)$. For which $p$ and $(X,d)$ is $(\mathcal{P}_p(...
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Conditions for existence of an optimal transport map between finitely supported probability measures
Let $E$ be a polish and let $P$ and $Q$ be finitely supported probability measures on $X$. What conditions are required to ensure that: for every $\delta>0$ there exists a $\delta$-optimal ...
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Wasserstein distance between $N(0,1/d)$ and the marginal distribution of $x_1$ when $x=(x_1,\ldots,x_d)$ is uniform on the unit-sphere in $R^d$
Let $x=(x_1,\ldots,x_d)$ be uniformly distributed on the unit-sphere in $\mathbb R^d$.
Question.
What is a good upper-bound for Wasserstein distance between $N(0,1/d)$ and the marginal distribution ...
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2
votes
1
answer
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How to prove the limit of minimizing sequence of measures is again absolutely continuous(w.r.t. Lebesgue) in the minimizing movement scheme?
I am considering the minimizing movement scheme related to the gradient of entropy functional in 2-Wasserstein space. The problem is to minimize the following functional for each fixed $\eta$ which is ...
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1
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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_{...
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2
votes
1
answer
222
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Given iid samples from the joint distribution $P$ of pair of r.v.'s $(X,Y)$, how to get iid samples from independence coupling $P_X \otimes P_Y$?
Let $(X,Y)$ be a pair of random variables on a measure space $\mathcal T \subseteq \text{"subsets of }\mathbb R^2\text{"}$, with joint probability distribution $P$.
We don't assume $X$ and $Y$ are ...
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0
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1
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Why is $\Xi \equiv 0$ if $E=C_{0}(X \times Y)$?
I'm reading the proof of Kantorovich duality from Villani's book Topics in Optimal Transportation. In page 28, the author said that
Exercise 1.11. Let us try to extend this proof to the non-compact ...
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3
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1
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403
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Regularity of transport map
Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}^n$ with first moment and suppose that both $\mu$ and $\nu$ have a densities with respect to the $n$-dimensional Lebesgue measure. Fix some ...
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3
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1
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529
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Wasserstein-type concentration inequalities for empirical measures on polish spaces
Let $(\mathcal{X},d)$ be a Polish (metric) space and let $\{X_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. $\mathcal{X}$-valued random elements defined on a common complete (standard) probability space ...
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1
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0
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193
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What is the role of of continuity in this proof of Kantorovich duality?
I'm reading the proof of Kantorovich duality from Villani's book Topics in Optimal Transportation.
Let $X$ and $Y$ be Polish spaces. Let $P(X), P(Y)$ be the spaces of all Borel probability measures ...
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0
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55
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"More" cyclical monotonicity
Let $X$ and $Y$ be some finite sets. For a given function $f:X\times Y\rightarrow \mathbb{R}$, we say a set $S\subset X\times Y$ is $f$-cyclically monotone if for any sequence $(x_1,y_1),...,(x_n,y_n)\...
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2
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201
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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 $\mu$ and $\nu$ on a metric space $X$ by
$$
W_p(\...
- 123
3
votes
1
answer
310
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Open problem 1.28: $W^{1,1}$ regularity for optimal transport map
While I don't work on the regularity theory for the optimal transport map, I was curious about the open problem 1.28 listed in Ambrosio and Gigli's User's guide: the problem to determine whether we ...
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vote
1
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261
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Parameterization of exponential family
Let $\{\mathbb{P}_{\theta}\}_{\theta}$ be an exponential family of probability measures, all with finite mean. Under what conditions is the parameterization map $\theta\mapsto \mathbb{P}_{\theta}$ ...
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8
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2
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741
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How does a statistical divergence change under a Lipschitz push-forward map?
Let $\mu, \nu$ be two probability measures on a space $X$ (assume Polish space).
$T: X \rightarrow Y$ is a Lipschitz-map that acts as a push-forward on these measures; let $\mu^\prime = T_{\#\mu}$ and ...
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0
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70
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$c$-cyclical monotonicity: does this proof hold if $f \equiv +\infty$ or $\int c \mathrm d \gamma = +\infty$?
I'm reading the proof of Theorem 1.38. from section 1.6.2 $c$-cyclical monotonicity and duality of Santambrogio's Optimal transport for applied mathematicians.
My understanding: It seems for the ...
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8
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1
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726
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First variation in $L^2$ sense of the square of the Wasserstein metric
Let me consider the functional $\mathcal{F}(\rho)=\mathcal{W}_2^2(\mu,\rho)$ defined in in the space of absolutely continuous probability measures $\mathcal{P}_{ac,2}(\Omega)$, where $\mathcal{W}_2^2$ ...
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256
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Sobolev variant of Wasserstein space
Let $\mathcal{P}(\mathbb{R}^n)$ be the set of Borel probability measures on the Euclidean space $\mathbb{R}^n$ and consider thereof consisting of all probability measures $\mathbb{P}$ satisfying $\int\...
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2
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0
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Estimate of Wasserstein distance and flow of vector fields under particular assumptions
Let $\mu$ be a compactly supported absolutely continuous probability measure. Let $v,u$ be Lipschitz vector fields. For a vector field $w$ recall that $\Phi_t^w$ denotes its flow.
A classical estimate ...
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3
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2
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336
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Invertibility of neural network as operator on Wasserstein space
Question statement: Consider the space of probability measures with finite second moments $P_2(\mathbb{R}^d)$, which is equipped with the Wasserstein-2 distance $W_2$, and the square integrable ...
- 1,127
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1
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406
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Computing discrete optimal transport
I am trying to find a combinatorial approach to solve the following optimization problem.
\begin{align}
&\max_{x_{ij}} C_{ij} x_{ij}, \\
&\text{such that},\\
&\sum_{j} x_{ij} \leq r_i~\...
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1
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1
answer
1k
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Closed-form upper-bounds for Wasserstein distance between finite measures
Let $x_1,\dots,x_n,y_1,\dots,y_n\in \mathbb{R}$ and such that $x_i\neq x_j$ and $y_i\neq y_j$ if $i\neq j$. Let $a,b$ be elements of the probability n-simplex. Define the measures $\mu\triangleq \...
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3
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1
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988
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About the metrizability of the space of Probability measures $\mathcal{P}(S)$
It is often proved in Books that the space of Probability measures $\mathcal{P}(S)$ on a Polish metric space $(S,\rho)$ endowed with the weak/narrow topology induced by declaring it to be be the ...
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1
vote
0
answers
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Transport-type duality for preduals of $C^{k,1}$-functions
Let $\Omega$ be a non-empty, simply connected, and open subset of $\mathbb{R}^d$ for some positive integer $d$. Let $k$ be a non-negative integer. Consider the Banach space $C^{k,1}_0(\Omega)$ ...
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1
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0
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71
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Hyperplanes which equalize the Radon transforms of two distributions
Let $p_1$ and $p_2$ be "nice" probability densities on $\mathbb R^m$, for example the densities of a multivariate Gaussians $N(\mu_1,\Sigma)$ and $N(\mu_2,\Sigma)$ with common covariance ...
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Eigenvalues of Witten Laplacian induced by log-concave probability measure on manifold
Let $M$ be a closed $n$-dimensional Riemannian manifold and let $\mu=e^{-V}d\mathrm{vol}_M$ be a log-concave probability measure on $M$, such that the pair $(M,\mu)$ verifies the so-called Bakry-Emery ...
- 6,853
4
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1
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Is the optimal transport of radially symmetric measures also radially symmetric?
Let $\mu$ and $\nu$ be radially symmetric probability measures on $\mathbb R^d$. Consider the Kantorovich optimal transport problem between $\mu$ and $\nu$, with convex, nonnegative cost. Suppose ...
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2
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1
answer
767
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Integer solution of optimal transport
Let us consider two vectors $\mathbf{a}=(a_1,...,a_n)$ and $\mathbf{b}=(b_1,...,b_m)$ so that each quantity is an integer $a_i,b_j \in \mathbb{N}$. It represents for example supply and demand. Let $\...
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1
answer
6k
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1-Wasserstein distance between two multivariate normal
The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by
$$d_p(\nu_{1},\nu_{2})=\left(\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,...
- 1,256
1
vote
1
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Expectation of a function according to a family of distributions
Consider a family of smooth, atomless CDFs, $F_x(\cdot)$, for each $x \in \mathbb R$. Suppose that $F_x(\cdot)$ are FOSD ranked in $x$. That is, for any $x, x'$ such that $x \ge x'$, $F_x(\cdot) \le ...
- 553
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0
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302
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Simplify Kantorovich–Rubinstein duality when distributions share a common marginal
Consider the product of two metric spaces $X\times Y$, and two probability distributions $\mu$ and $\nu$ on this product space. By the Kantorovich-Rubinstein duality, I can write the Wasserstein-1-...
- 53
1
vote
1
answer
100
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$L^p$-barycenters via continuous selectors
Let $\mathbb{P}$ be a Borel probability measure on $\mathbb{R}^n$ with $\int_{x \in \mathbb{R}^n} \|x\|^p\mathbb{P}(dx)<\infty$. For which $p\in [1,\infty)$ (other than $p=2$) is the following ...
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7
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1
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Stability of displacement interpolation in optimal transport
Let $(X,d)$ be a complete separable metric space, and let $(\mathcal{P}_2 (X), W_2)$ be the space of probability measures on $X$ with finite second moments, equipped with the 2-Wasserstein distance. ...
- 660
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2
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2k
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Wasserstein distance and the Kantorovich-Rubinstein duality
The only few references I could find on this topic are either amateur blog posts (http://n.ethz.ch/~gbasso/download/A%20Hitchhikers%20guide%20to%20Wasserstein/A%20Hitchhikers%20guide%20to%...
- 2,246
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1
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Is there an analogue of transportation-cost inequality under a weighted Log-Sobolev Inequality?
It is known that under the Log-Sobolev Inequality for $\pi$, i.e., if for all $\rho$,
$$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{\...
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One question about how to get $\inf_{\nu\in\mathcal{M}(Z)} \sum_{j=0}^N F_j^*(\nu)=-(\sum_{j=0}^N F_j^*)^*(0)$?
In this paper: Matching for teams by G. Carlier and I. Ekeland (2010), https://www.jstor.org/stable/25619994?seq=1#metadata_info_tab_contents: they claim that follows a standard argument in Ekeland ...
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votes
2
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2k
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The geometric median of a triangle
Let $\Omega\subset \mathbb R^n$ be a compact domain of dimension $n$. Define the geometric median on $\Omega$ as the point $m_{\Omega}\in \mathbb R^n$ such that the integral $\int_{\Omega}|x-m_{\Omega}...
- 14.3k
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1
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275
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Upper bound on the $p$-Wasserstein distance $\mathcal{W}_p(\xi,a\,\xi)$ for some constant $a \neq 0$
Let $\xi$ be a random vector taking values in $\mathbb{R}^d$. Is there an upper bound on the $p$-Wasserstein distance $\mathcal{W}_p(\xi,a\,\xi)$ for some constant $a \neq 0$?
I have seen that if $p=...
4
votes
0
answers
117
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Improving log-Sobolev inequalities via quadratic regularisation
Suppose that $\mu(dx) = \exp(-\psi(x)) \, \mathrm{dx}$ is a probability measure on $\mathbf{R}^d$.
For suitable functions $g \geqslant 0$, define
$$\text{Ent}(g) = \int \mu(dx) g(x) \log \left( \frac{...
- 801
1
vote
1
answer
241
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Continuity of pushforward operation
Let $X$ and $Y$ be compact metric spaces and let $f,g:X\rightarrow Y$ be $\epsilon$-uniformly close; i.e.:
$$
\sup_{x \in X} d_Y(f(x),g(x))<\epsilon.
$$
Then, are their push-forwards close in ...
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