# Questions tagged [optimal-transportation]

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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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### 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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### 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 ...
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### Moduli of continuity and Wasserstein differentiability of functions between measures

Let $X=\mathbb{R}^n$; I am also interested in the general case $X$ is a metric space but for simplicity let's focus on Euclidean space. Let $\mathcal{P}(X)$ denote the space of Borel probability ...
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Let $p>1$ and $\mu\neq \nu$ be two probability measures on $\Omega\subset \mathbb{R}^d$ a bounded set. For $\alpha \geq 0$, we let $$C_\alpha(\mu,\nu) = \inf_\sigma \frac{W_p(\mu+\sigma,\nu+\sigma)}... 1answer 105 views ### Ideas on how to prove Pythagorean identity involving Wasserstein distances? I conjectured earlier that if P and Q were two probability measures, then we could show$$W^2(P,Q) = \min_{T} [d^2(P,T_{\#}P) + W^2(T_{\#}P,Q)]$$where W^2(P,Q) denotes the squared Wasserstein-2 ... 1answer 265 views ### 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. ... 2answers 706 views ### Do distance functionals separate probability measures? Let (\Omega,d) be a compact metric space and \mathcal P(\Omega) its space of Borel probability measures. Let D=\{ d_p\mid p\in\Omega\} where d_p(x)=d(p,x) be the set of all "distance ... 1answer 85 views ### Wasserstein space with strictly non-positive sectional curvature Let (X,d) be a (separable, complete) metric space with uniformly strictly non-positive curvature in the sense of Alexandrov, i.e. (X,d) satisfies a CAT(K) inequality for some K<0. Does it ... 1answer 161 views ### 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 ... 2answers 102 views ### Are there alternative regularizations for optimal transport problems besides entropic regularization? I see that most of the regularization done involves an entropy term. Has there been any work done on other regularization methods? In particular, I'm wondering if anyone has done a regularization ... 0answers 89 views ### Upper bound \tau_C := \int_{\|x\| \le 1}(vol(C \cap (x + C))/vol(C))dx for a convex body C \subseteq \mathbb R^n, by reducing to a ball Let C be a convex body in \mathbb R^n, i.e a bounded convex subset of \mathbb R^n which has nonempty interior, and which is (A) open, or (B) closed (I'm not sure one makes more sense; choose the ... 0answers 63 views ### Metrics on the space of distributions in terms of p.d.fs If two probability distributions (on the same measure space) are s.t they have p.d.fs and the L^1 distance between the p.d.f.s is large, then is there a choice of a nice" metric d_{\rm ... 1answer 159 views ### Optimal transport: find cost function given observed transport Could you advise me please on what to read on the "inverse" problem: suppose I have a source measure, a target measure and I observe the solution to optimal transport problem -- can I "back out" the ... 1answer 200 views ### Heat flow, decay of the Fisher information, and \lambda-displacement convexity In the whole post I will work in the flat torus \mathbb T^d=\mathbb R^d/\mathbb Z^d and \rho will stand for any probability measure \mathcal P(\mathbb T^d). This question is strongly related to ... 0answers 69 views ### improved regularization for \lambda-convex gradient flows It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ... 0answers 75 views ### Explicit formula for this distance between positive semi-definite matrices? Let A and B in \mathbb{R}^{d\times d} be positive semi-definite (psd) matrices and let d\tau be the uniform probability distribution on the unit sphere \mathbb{S}^{d-1} in \mathbb{R}^d. I ... 0answers 141 views ### A new “adversarial” Wasserstein distance? Let us consider \mu_1, \mu_2 and \mu_3 three probability measures living on [0,1]^{k_1}, [0,1]^{k_2} and [0,1]^krespectively, with k_1 +k_2=k. Let us denote by \Gamma(\mu,\nu) the set of ... 1answer 114 views ### Is the Wasserstein kernel positive definite? Define a point cloud X=\{x_i\}_{1\leq i\leq n}, for x_i\in\mathbb R^d. Define the Wasserstein kernel as$$W(X,Y)=\max_{T}\frac{1}{n}\sum_{kl}T_{kl}\langle x_k,y_l \rangle$$where T is any doubly ... 0answers 78 views ### Prove the equicontinuity of a maximizing sequence Let X be a compact subset of \mathbb{R} and c(x_1,x_2,x_3,x_4) be a fixed bounded continuous functions on X^4. Assume \mu,\nu are probability measures on X^2, and \mu\otimes\nu is the ... 1answer 318 views ### Transport of measure Let's disintegrate \mu and \nu, two probabilities on \mathbb{R}^{d} , according to$$ \pi_{k} (x_{1},...,x_{d}) = (x_{k},...,x_{d}) We get a family of measures and each measure \mu_{k,d}^{+... 1answer 173 views ### 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~\... 1answer 78 views ### A problem with the dual form of semi-discrete optimal transport Consider the uniform distribution \lambda on [0,1], and a point measure \rho with density \frac{1}{2} (\delta_{x_1} + \delta_{x_2}), where we have 0\le x_1 \le x_2 < 1/2. If our cost is ... 0answers 171 views ### Variational derivative of Wasserstein distance using Benaumou-Brenier formulation I learned from the gradient flow theory in Wasserstein space that an equation of gradient flow type\partial_t \rho + \nabla \cdot (\rho \nabla \frac{\delta F}{\delta \rho})=0, can be derived as ...
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,...
Kantorovich's optimal transportation problem $$\tau_c(\mu,\nu)=\min\limits_{\pi\in\Pi(\nu,\mu)} \int_{X\times Y}c(x,y)d\pi(x,y)$$ where \$\Pi(\mu,\nu) = \{\pi\in P(X\times ...