# Questions tagged [optimal-transport]

For questions related to the theory, applications, and computational aspects of optimal transport and related topics such as the Wasserstein (and other transportation cost) distances, the Monge-Ampere equation, metric gradient flows, martingale optimal transport, and optimal matching.

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### Optimal objective of Static Schrodinger Bridge

I refer to notes about entropy-regularized optimal transport, at https://www.math.columbia.edu/~mnutz/docs/EOT_lecture_notes.pdf In Theorem 3.2, it says that the Schrodinger potentials achieve the ...
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### Derivative of Kantorovic Potential wrt to Measure

The Kantorovich Dual of the 1-Wasserstein distance $W_1(p,q)$ between two densities $p(x), q(x)$ is given by $$W_1(p,q) = \sup_{|f|_L\leq 1} \int f(x)(p(x)-q(x))dx$$ with $|f|_L$ denoting the ...
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### Computational optimal transport [closed]

Let's say I have two piles of sand with the same amount of grains, but a different shape. Is there a way to "optimally transport" pile A to B in N steps by moving a handful of grains each ...
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### Is the composition of two optimal transport maps still optimal (under some assumptions)?

Consider three absolutely continuous probability measures $\mu$, $u$, and $\nu$ on $\mathbb R^d$ ($d \geq 1$), all of which have finite second moments. A transport map from $\mu$ to $\nu$ is called ...
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1 vote
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### Embedding $X \ni x \mapsto \delta_x \in P_2(X)$ is totally convex

I am looking for a reference to a proof of the following result: Let $X$ be a compact, connected, smooth Riemannian manifold. Then, the embedding $$X \ni x \mapsto \delta_x \in P_2(X)$$ has totally ...
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### Existence of joint optimal coupling

Given probability distributions $(\mu_1, \mu_2,\dots,\mu_n)$ on a sufficiently nice space $X$, does there always exist a random vector $(X_1,X_2\dots,X_n)$ such that any pair $(X_i,X_j)$ is an optimal ...
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### Optimal Transport and Entropic Regularization

We are working with discrete optimal transport. Let $P$ be a matrix and let $H(P) =- \sum_{i,j} P_{i,j} (\log(P_{i,j})-1)$. Let $C$ be the cost matrix. And $\langle C,P\rangle$ the Frobenius inner ...
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### More examples for non-existent Monge maps

As stated very often, the Monge-Problem in Optimal Transport does not always admit a solution, for example when the start distribution $\mu$ is Dirac delta but the target distribution $\nu$ is not, ...
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1 vote
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### Change of variables - Wasserstein distance

My question is how exactly the change of variable formula is applied in the second row of the equation?
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### The composition of optimal transport maps is no longer an optimal transport map

Let $X,Y,Z$ be metric spaces. Let $\mu,\nu,\omega$ be the probability measures on $X,Y,Z$, respectively. Moreover, assume all three measures vanish on small sets. Assume $T:X\to Y$ is the optimal ...
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### Geometric interpretation of barycenter in Wasserstein-2 space

Given $\mathcal{X} \subset \mathbb{R}^d$. Let $\mu_k=\frac{1}{n}\sum_{i=1}^n x_{ki}$ for $k=1,\cdots, K$ be probability measures on $\mathcal{X}$. The Wasserstein-2 barycenter $\mu_1,\cdots, \mu_K$ is ...
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### Composition of two transport maps and pushforward operator

Given two absolutely continuous probability measures $\mu,\sigma \in \mathcal P_2(\mathbb R^n)$ and two maps $T_1, T_2$ such that $$(T_1 \circ T_2)_\#\sigma =\mu$$ where $(\cdot)_{\#}$ denotes the ...
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### Partial order on matrix

In Barycenters in the Wasserstein space https://hal.archives-ouvertes.fr/hal-00637399/document section 6.3 The gaussian case Theorem 6.1, the authors used the order on the matrix ring by claiming the ...
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