Optimal transport problems regularized by generic convex functions: a geometric and algorithmic approach

We present a method to efficiently compute Wasserstein gradient flows. Our approach is based on a generalization of the back-and-forth method (BFM) introduced in Jacobs and Léger [Numer. Math. 146 (2020) 513–544.]. to solve optimal transport problems. We evolve the gradient flow by solving the dual problem to the JKO scheme. In general, the dual problem is much better behaved than the primal problem. This allows us to efficiently run large scale gradient flows simulations for a large class of internal energies including singular and non-convex energies.

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Quantitative Stability in the Geometry of Semi-discrete Optimal Transport

International Mathematics Research Notices ◽

10.1093/imrn/rnaa355 ◽

2020 ◽

Author(s):

Mohit Bansil ◽

Jun Kitagawa

Keyword(s):

Optimal Transport ◽

Convex Geometry ◽

Regularity Theory ◽

Uniform Norm ◽

Quantitative Stability ◽

Wirtinger Inequality ◽

Dual Variables ◽

Transport Problems ◽

Monge Ampere ◽

Stability Results

Abstract We show quantitative stability results for the geometric “cells” arising in semi-discrete optimal transport problems. We first show stability of the associated Laguerre cells in measure, without any connectedness or regularity assumptions on the source measure. Next we show quantitative invertibility of the map taking dual variables to the measures of Laguerre cells, under a Poincarè-Wirtinger inequality. Combined with a regularity assumption equivalent to the Ma–Trudinger–Wang conditions of regularity in Monge-Ampère, this invertibility leads to stability of Laguerre cells in Hausdorff measure and also stability in the uniform norm of the dual potential functions, all stability results come with explicit quantitative bounds. Our methods utilize a combination of graph theory, convex geometry, and Monge-Ampère regularity theory.

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Estimating matching affinity matrices under low-rank constraints

Information and Inference A Journal of the IMA ◽

10.1093/imaiai/iaz015 ◽

2019 ◽

Vol 8 (4) ◽

pp. 677-689

Author(s):

Arnaud Dupuy ◽

Alfred Galichon ◽

Yifei Sun

Keyword(s):

Joint Distribution ◽

Transport Theory ◽

Optimal Transport ◽

Low Rank ◽

Rank Matrix ◽

Entropic Regularization ◽

Optimal Transport Theory ◽

Rank Constraints ◽

Transport Problems ◽

Low Rank Matrix

Abstract In this paper, we address the problem of estimating transport surplus (a.k.a. matching affinity) in high-dimensional optimal transport problems. Classical optimal transport theory specifies the matching affinity and determines the optimal joint distribution. In contrast, we study the inverse problem of estimating matching affinity based on the observation of the joint distribution, using an entropic regularization of the problem. To accommodate high dimensionality of the data, we propose a novel method that incorporates a nuclear norm regularization that effectively enforces a rank constraint on the affinity matrix. The low-rank matrix estimated in this way reveals the main factors that are relevant for matching.

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Entropic regularization of continuous optimal transport problems

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2020.124432 ◽

2021 ◽

Vol 494 (1) ◽

pp. 124432 ◽

Cited By ~ 1

Author(s):

Christian Clason ◽

Dirk A. Lorenz ◽

Hinrich Mahler ◽

Benedikt Wirth

Keyword(s):

Optimal Transport ◽

Entropic Regularization ◽

Transport Problems

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