Entropic optimal transport: convergence of potentials
By Marcel Nutz, Johannes Wiesel. Probability theory and related fields, 2021
101 citations recorded by the source corpus.
We study the potential functions that determine the optimal density for ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}-entropically regularized optimal transport, the so-called Schrödinger potentials, and their convergence to the counterparts in classical optimal transport, the Kantorovich potentials. In the limit ε→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} of vanishing regularization, strong compactness holds in L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{1}$$\end{document} and cluster points are Kantorovich potentials. In particular, the Schrödinger potentials converge in L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{1}$$\end{document} to the Kantorovich potentials as soon as the latter are unique. These results are proved for all continuous, integrable cost functions on Polish spaces. In the language of Schrödinger bridges, the limit corresponds to the small-noise regime.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/bf7912c6e87af22a195819b8a8faa578c97931b4
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities (2022, methodology): “Upon passing to a sequence εk ց 0, existing results on the convergence of entropic optimal transport potentials show that φεk → φ0 in L(1)(P ) (see [NW21]).”