# Questions tagged [optimal-transportation]

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169
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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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### 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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### 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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### Fokker-Planck: equivalence between linear spectral gap and nonlinear displacement convexity?

In a smooth, bounded and convex domain $\Omega\subset \mathbb R^d$ consider the usual linear Fokker-Planck equation with Neumann (some would say Robin) boundary conditions
\begin{equation}
\label{FP}
\...

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

Let $(X, | \cdot |)$ be a Banach space.
I am investigating whether one can extend the definition of the Kullback-Leibler divergence
$$
\text{KL}(\mu \ \Vert \ \nu)
:= \int_{\Omega} \ln\left(\frac{\...

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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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### Hyperbolic space embeds into Wasserstein space

Fix a positive integer $n$, let $\mathbb{H}^n$ be the $n$-dimensional hyperbolic space, $r>0$, $x\in \mathbb{H}^n$ and consider the closed (compact) geodesic ball $B_{\mathbb{H}^n}(x,r)$. Are ...

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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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### 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 ...

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### Measurability of Markov kernel wrt the Borel $\sigma$-algebra generated by the weak topology

Consider two Polish metric probability spaces $(\mathcal{A}, \Sigma_\mathcal{A})$ and $(\mathcal{B}, \Sigma_\mathcal{B})$, endowed with their Borel $\sigma$-algebras. Denote as $\mathcal{P}_\mathcal{B}...

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### Sufficient condition for an $n$-tuple to be a convex conjugate

We say $(f_1,f_2,\dotsc,f_N)$ is a convex conjugate if for any $i=1,2,\dotsc,N$ and any $x_i\in\Bbb R^d$, we have:
$$f_i(x_i)=\sup\left\{\sum_{k=1}^{N}\sum_{j=k+1}^N x_k x_j - \sum_{j=1,j\neq i}^N f_j(...

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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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### Nontrivial lower-bound for $\mathbb P(f(X) \le \alpha \|\nabla f(X)\|)$

Let $f:\mathbb R^d \to \mathbb R$ be a differentiable function whose gradient is $L$-Lipschitz (w.r.t euclidean norm), for some fixed $L \in [0,\infty)$. Let $X=(X_1,\ldots,X_d)$ be a concentrated ...

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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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### 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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### 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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### 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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### Reference request: optimal transport from Wiener to Poisson

This question arises to me when reading the paper OPTIMAL TRANSPORT FROM LEBESGUE TO POISSON : Let $\mathcal D(\mathbb R_+)$ be the space of right-continuous functions $x:\mathbb R_+\to\mathbb R$ s.t. ...

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144
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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 ...

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### Uniqueness of maximizer of dual Kantorovich problem with quadratic(or any strictly convex) cost

I am considering the optimal transport problem under the setting $X=\mathbb{R}^n$, $\mu,\nu\in\mathcal{P}(X)$ be two probability measures, and the cost function is $c(x,y)=|x-y|^2$. We know from ...

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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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### 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(\...

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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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### Bounding the Jacobian determinant on the mapping given by Brenier’s Theorem

I have two distributions with small 2-Wasserstein distance between them that satisfy the conditions for Brenier's theorem. I am looking for conditions under which I can prove $p(x)\leq(1+a)\cdot q(x)+...

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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{...

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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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### 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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### Empirical estimation of Brenier map from data

Let $f:\mathbb R^d \to \mathbb R$ be a "nice" (say, continuous) function define $A = A_f := \{x \in \mathbb R^d \mid f(x) \ge 0\}$ and $B =B_f:= \{x \in \mathbb R^d \mid f(x) \le 0\}$, and ...

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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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### $L^p$-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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### Textbooks or lecture notes about mean field games

I am looking for a good introductory level textbook (or lecture notes) on mean field games that would be suitable for a graduate course. Ideally, it would include some brief words about optimal ...

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### What is the intuition behind the Kantorovich potential in optimal transport?

From what I currently understand, under certain conditions one may turn the usual Kantorovich problem - a minimisation problem in terms of measures into a maximisation problem in terms of functions. ...

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### Convexity of a set of probability densities

Consider the space of probability densities $(P(\mathbb{R}^d), W_2)$ (probability measures on $\mathbb{R}^d$ with 2-Wasserstein distance).
How can we determine if a subset $Q$ is convex?
I know that a ...

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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=...

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### Orthogonality in Wasserstein tangent space for discrete measures with equal mass

Let say I have $N$ discrete probability measures $(\mu_1,...,\mu_N)$ where each of them has $n$ points in $\mathbb{R}^2$ of equal mass.
Let $P(\mathcal{X})$ be the space of these probability measures ...

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### Reweighting probability measures by convex potentials, and contraction in transport distance

Let $W: \mathbf{R}^d \to \mathbf{R}$ be a convex function such that $\int \exp(-W) = 1$, and define probability measures $\mu_y$ by
$$\mu_y (dx) = \exp( - W (x - y)) \,dx,$$
i.e. each $\mu_y$ is a ...

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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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### Closure of finite support measures in the Wasserstein metric

This is a follow-up question to this question:
Closure Wasserstein for pointmasses
Let $(X,d)$ be a metric space, and let $W_1(X)$ be the space of probability measures $\mu$ on $X$ having finite first ...

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### Producing a minimiser for the Kantorovich problem from a minimiser of the Beckmann flow problem

Notation: We denote by $\mathcal M$ the set of vector valued measures on $\mathbb R^d$ whose divergence is a scalar measure (in the weak sense).
Definitions: Consider the Beckmann flow minimisation ...

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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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### Is integration against an indicator Wasserstein-Continuous

Let $\mathcal{P}_p(X)$ denote the Wasserstein space over a compact metric space $X$, and $1\leq p<\infty$. Fix a non-empty closed subset $C\subseteq X$. Then is the map:
$$
\mathbb{P} \mapsto \...

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### Semiconcavity estimate for the squared distance on a compact Riemannian manifold

I am currently reading this paper on the Riemannian structure of the Wasserstein space over a compact Riemannian manifold (my question doesn't concern the Wasserstein metric), specifically Section 4.1,...

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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-...

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### 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^p(\mu, \nu) = \inf_{\gamma \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^p\; d\...

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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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### 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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### How does one define weak convergence of probability measures in $L^{\infty}(\Omega)$?

I am reading the following article and on page 9/17 (above Eqn (4.9)) the authors state that if $\gamma_{\epsilon_k}|\_G_{\delta}\times \Omega\to \gamma|\_G_{\delta}\times \Omega$ as $\epsilon_k\to 0$ ...

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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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### Arbitrarily bad rates of convergence in Wasserstein metric

Suppose $W_p(\mu_n,\mu)\to 0$ and $d(E(\mu_n),E(\mu))<r_n$. Here, $W_p$ is the $p$th-order Wasserstein distance (with respect to the metric $d$) and $\mu_n,\mu$ are probability measures on some ...

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### How to interpret couplings in optimal transport?

Let $\mu$ and $\nu$ be two measures on some (at least measurable) space $X$. In optimal transport theory, Monge's problem to
$$ \text{minimize} \quad \int c(x,T(x))\mu(dx) \quad \text{over measurable ...