scholarly journals Entropic regularization of continuous optimal transport problems

2021 ◽  
Vol 494 (1) ◽  
pp. 124432 ◽  
Author(s):  
Christian Clason ◽  
Dirk A. Lorenz ◽  
Hinrich Mahler ◽  
Benedikt Wirth
Keyword(s):  
Optimal Transport ◽  
Entropic Regularization ◽  
Transport Problems

Estimating matching affinity matrices under low-rank constraints

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.

scholarly journals A tumor growth model of Hele-Shaw type as a gradient flow

2020 ◽  
Vol 26 ◽  
pp. 103
Author(s):  
Simone Di Marino ◽  
Lénaïc Chizat
Keyword(s):  
Tumor Growth ◽  
Optimal Transport ◽  
Gradient Flow ◽  
Convex Domains ◽  
Wasserstein Metric ◽  
Uniqueness Of Solutions ◽  
Tumor Growth Model ◽  
Entropic Regularization ◽  
Minimizing Movements ◽  
Convergence Of The Scheme

In this paper, we characterize a degenerate PDE as the gradient flow in the space of nonnegative measures endowed with an optimal transport-growth metric. The PDE of concern, of Hele-Shaw type, was introduced by Perthame et. al. as a mechanical model for tumor growth and the metric was introduced recently in several articles as the analogue of the Wasserstein metric for nonnegative measures. We show existence of solutions using minimizing movements and show uniqueness of solutions on convex domains by proving the Evolutional Variational Inequality. Our analysis does not require any regularity assumption on the initial condition. We also derive a numerical scheme based on the discretization of the gradient flow and the idea of entropic regularization. We assess the convergence of the scheme on explicit solutions. In doing this analysis, we prove several new properties of the optimal transport-growth metric, which generally have a known counterpart for the Wasserstein metric.

scholarly journals Scaling algorithms for unbalanced optimal transport problems

2018 ◽  
Vol 87 (314) ◽  
pp. 2563-2609 ◽  
Author(s):  
Lénaïc Chizat ◽  
Gabriel Peyré ◽  
Bernhard Schmitzer ◽  
François-Xavier Vialard
Keyword(s):  
Optimal Transport ◽  
Transport Problems ◽  
Scaling Algorithms

scholarly journals Approximation of optimal transport problems with marginal moments constraints

2020 ◽  
Vol 90 (328) ◽  
pp. 689-737
Author(s):  
Aurélien Alfonsi ◽  
Rafaël Coyaud ◽  
Virginie Ehrlacher ◽  
Damiano Lombardi
Keyword(s):  
Optimal Transport ◽  
Transport Problems

scholarly journals The back-and-forth method for Wasserstein gradient flows

2021 ◽  
Vol 27 ◽  
pp. 28
Author(s):  
Matt Jacobs ◽  
Wonjun Lee ◽  
Flavien Léger
Keyword(s):  
Large Class ◽  
Large Scale ◽  
Dual Problem ◽  
Optimal Transport ◽  
Gradient Flow ◽  
Gradient Flows ◽  
Primal Problem ◽  
Transport Problems ◽  
Wasserstein Gradient Flows ◽  
Numer Math

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.

scholarly journals Quantitative Stability in the Geometry of Semi-discrete Optimal Transport

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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