Convergence Rates for Regularized Optimal Transport via Quantization

2022/08/30 by Stephan Eckstein, Marcel Nutz, Eckstein, Stephan +1 · 3 citations
Mathematics · #Geometric Analysis and Curvature Flows #Markov Chains and Monte Carlo Methods #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2208.14391

Abstract

We study the convergence of divergence-regularized optimal transport as the regularization parameter vanishes. Sharp rates for general divergences including relative entropy or L^p regularization, general transport costs and multi-marginal problems are obtained. A novel methodology using quantization and martingale couplings is suitable for non-compact marginals and achieves, in particular, the sharp leading-order term of entropically regularized 2-Wasserstein distance for all marginals with finite (2+δ)-moment.

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• General Divergence Regularized Optimal Transport: Sample Complexity and Central Limit Theorems
• Error estimates for deterministic empirical approximations of probability measures
• Linear Convergence of Gradient Descent for Quadratically Regularized Optimal Transport

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