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# Questions tagged [optimal-transportation]

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### Is this map (from the space of probability densities to the Wasserstein space) Lipschitz?

Let $p \in [1, \infty)$. Let $\mathcal P_p(\mathbb R^d)$ be the space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moments. Let $D_p$ be the collection of all Borel measurable ...
1 vote
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### Strict convexity of the cost function is enough to ensure the existence and uniqueness of the optimal transport map

Let $X=Y = \mathbb R^d$ and $c:X \times Y \to [0, \infty)$ be Borel measurable. Let $\mu, \nu$ be Borel probability measures on $X,Y$ respectively. Let $\mathcal T$ be the set of all Borel measurable ...
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### Optimal Transport: how is this transport map Borel measurable?

I'm reading Theorem 1.17. and its proof at page 14 of Santambrogio's Optimal transport for applied mathematicians. The content is not hard but a little bit long (because of related detail). Please ...
1 vote
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### Continuity of the traffic intensity

Note: We say that a Borel measure on $\mathbb R^n$ has a continuous density if it is absolutely continuous with respect to Lebesgue measure and has a continuous Radon Nikodym derivative. Let $\mu$ be ...
1 vote
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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 ...
1 vote
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### What are the "applications" of quantum optimal transport?

A quantum version of the Monge-Kantorovich optimal transport problem aims at optimizing a Hermitian cost matrix $C$ over the set of all bipartite coupling states $\rho_{AB}$, s.t. both of its reduced ... 190 views

### Optimal transport for applied mathematicians: how does $\varphi (x) = \inf_{y \in Y} [c(x, y) - \psi (y)] \neq -\infty$ follow in Theorem 1.37?

I'm reading a proof of Theorem 1.37 from Santambrogio's Optimal transport for applied mathematicians: calculus of variations, PDEs, and modeling. First, I quote related definitions. Let $X,Y$ be ...
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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 $\mu$?
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### Does $\mathscr{H}^{d-1} (A)<+\infty$ for $A\subset \mathbb R^d$ imply $A$ is (Borel) measurable?

I'm reading section 1.3.1 The quadratic case in $\mathbb{R}^{d}$ at page 17 from Santambrogio's Optimal Transport for Applied Mathematicians. The PDF is freely available from here. Let $\mu$ be a ...
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### Uniqueness of Kantorovich potentials?

$\newcommand{\R}{\mathbb R}$Take $\Omega\subset \R^d$ bounded, convex, and smooth. Consider the optimal transport problem with cost $c(x,y)=\lvert x-y\rvert^2$, leading to the quadratic Wassersein ...
1 vote
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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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### 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 ...
1 vote
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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 ...
1 vote
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 $\int\... 4 votes 1 answer 277 views ### Fokker-Planck: equivalence between linear spectral gap and nonlinear displacement convexity? In a smooth, bounded and convex domain$\Omega\subset \mathbb R^d$consider the usual linear Fokker-Planck equation with Neumann (some would say Robin) boundary conditions \begin{equation} \label{FP} \... 3 votes 0 answers 222 views ### Radon-Nikodym derivative of vector-valued measure with respect to another vector-valued measure Let$(X, | \cdot |)$be a Banach space. I am interested in whether one can extend the definition of the Kullback-Leibler divergence $$\text{KL}(\mu \ \Vert \ \nu) := \int_{\Omega} \ln\left(\frac{\... 2 votes 1 answer 131 views ### Given iid samples from the joint distribution P of pair of r.v.'s (X,Y), how to get iid samples from independence coupling P_X \otimes P_Y? Let (X,Y) be a pair of random variables on a measure space \mathcal T \subseteq \text{"subsets of }\mathbb R^2\text{"}, with joint probability distribution P. We don't assume X and Y are ... 7 votes 3 answers 298 views ### Hyperbolic space embeds into Wasserstein space Fix a positive integer n, let \mathbb{H}^n be the n-dimensional hyperbolic space, r>0, x\in \mathbb{H}^n and consider the closed (compact) geodesic ball B_{\mathbb{H}^n}(x,r). Are ... 1 vote 0 answers 60 views ### Hyperplanes which equalize the Radon transforms of two distributions Let p_1 and p_2 be "nice" probability densities on \mathbb R^m, for example the densities of a multivariate Gaussians N(\mu_1,\Sigma) and N(\mu_2,\Sigma) with common covariance ... 1 vote 1 answer 51 views ### Expectation of a function according to a family of distributions Consider a family of smooth, atomless CDFs, F_x(\cdot), for each x \in \mathbb R. Suppose that F_x(\cdot) are FOSD ranked in x. That is, for any x, x' such that x \ge x', F_x(\cdot) \le ... 0 votes 1 answer 199 views ### Measurability of Markov kernel wrt the Borel \sigma-algebra generated by the weak topology Consider two Polish metric probability spaces (\mathcal{A}, \Sigma_\mathcal{A}) and (\mathcal{B}, \Sigma_\mathcal{B}), endowed with their Borel \sigma-algebras. Denote as \mathcal{P}_\mathcal{B}... 1 vote 0 answers 32 views ### Sufficient condition for an n-tuple to be a convex conjugate We say (f_1,f_2,\dotsc,f_N) is a convex conjugate if for any i=1,2,\dotsc,N and any x_i\in\Bbb R^d, we have:$$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(... 0 votes 2 answers 185 views ### 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$...
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 ...