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Convergence rate of general entropic optimal transport costs

Carlier, Guillaume; Pegon, Paul; Tamanini, Luca (2022), Convergence rate of general entropic optimal transport costs. https://basepub.dauphine.psl.eu/handle/123456789/23106

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Type
Document de travail / Working paper
Date
2022
Series title
Cahier de recherche CEREMADE, Université Paris Dauphine-PSL
Published in
Paris
Pages
31
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Author(s)
Carlier, Guillaume
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Pegon, Paul
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Tamanini, Luca
Bocconi Institute for Data Science and Analytics [BIDSA]
Abstract (EN)
We investigate the convergence rate of the optimal entropic cost vε to the optimal transport cost as the noise parameter ε↓0. We show that for a large class of cost functions c on Rd×Rd (for which optimal plans are not necessarily unique or induced by a transport map) and compactly supported and L᪲ marginals, one has vε − v0 = d/2 εlog(1/ε) + O(ε). Upper bounds are obtained by a block approximation strategy and an integral variant of Alexandrov's theorem. Under an infinitesimal twist condition on c, i.e. invertibility of ∇²xy c, we get the lower bound by establishing a quadratic detachment of the duality gap in d dimensions thanks to Minty's trick.
Subjects / Keywords
Optimal transport; Entropic regularization; Schrödinger problem; Convex analysis; Entropy dimension.