Entropic approximation of \(\infty \)-optimal transport problems
type: Item
instance of: scholarly article
title: Entropic approximation of \(\infty \)-optimal transport problems
author: Camilla Brizzi
author: Guillaume Carlier
author: Luigi De Pascale
published in: Applied Mathematics and Optimization
publication date: 2024-08-20T00:00:00Z
review text: The paper presents an approach to solving \(\infty\)-optimal transportation problems with a supremal cost of the form:\N\[\N\inf_{\gamma \in \Pi(\mu,\nu)} \gamma - \operatorname{ess sup} c = \|c\|_{L^{\infty}(\gamma)}.\N\]\NThe authors propose an entropic penalisation method for approximating \(L^\infty\) optimal transport problems.via introducing a functional \(J_{p,\epsilon}\) which combines the transport cost raised to the power of \(p\) and an entropic term. \(\Gamma\)-convergence of \(J_{p,\epsilon}\) to \(J_\infty\) under certain conditions on \(\epsilon\) and \(p\) is established, which demonstrates that, in the limit case, \(\infty\)-cyclic monotone plans are selected.\N\NThe authors also give error estimates in the form of an upper bound on the speed of convergence of \(v_p = \min_{\gamma \in \Pi(\mu,\nu)} J_p\) to \(v_\infty = \min_{\gamma \in \Pi(\mu,\nu)} J_p\).\N\NIn the discrete case, these error estimates are sharpened by also providing lower bounds on the speed of convergence.\N\NFinally, numerical illustrations using the Sinkhorn algorithm are presented based on the Sinkhorn algorithm to solve the entropically penalised transport problems. This demonstrates the convergence of the transport plans to \(\infty\)-cyclically monotone plans, and illustrates that the convergence speed of \(v_p\) to \(v_\infty\) adheres with its theoretical bounds.
reviewed by: Dimitris P. Vartziotis
Mathematics Subject Classification ID: 49Q22
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 7898799
zbMATH Keywords: \(\infty\)-optimal transport
zbMATH Keywords: \(\infty\)-cyclical monotonicity
zbMATH Keywords: entropic approximation
MaRDI profile type: Publication
cites work: \({\mathcal{W}}_\infty \)-transport with discrete target as a combinatorial matching problem
cites work: Optimal transportation, modelling and numerical simulation
cites work: Convergence of Entropic Schemes for Optimal Transport and Gradient Flows
cites work: The $\infty$-Wasserstein Distance: Local Solutions and Existence of Optimal Transport Maps
cites work: The geometry of optimal transportation
cites work: The \(L^\infty\) optimal transport: infinite cyclical monotonicity and the existence of optimal transport maps
cites work: From the Schrödinger problem to the Monge-Kantorovich problem
cites work: Optimal transport: discretization and algorithms
cites work: Computational Optimal Transport: With Applications to Data Science
cites work: Optimal transport for applied mathematicians. Calculus of variations, PDEs, and modeling
cites work: Q4805362
cites work: Optimal Transport
DOI: 10.1007/S00245-024-10136-3
description: scientific article; zbMATH DE number 7898799