Convergence of Batch Greenkhorn for Regularized Multimarginal Optimal Transport

TL;DR

This paper introduces a batch Greenkhorn algorithm for multimarginal regularized optimal transport, providing convergence guarantees and extending existing algorithms like Sinkhorn and MultiSinkhorn with improved theoretical insights.

Contribution

It proposes a general batch Greenkhorn method with convergence analysis, unifying and extending prior algorithms for multimarginal optimal transport.

Findings

Global linear convergence rate established

Explicit iteration complexity bounds derived

Extensions improve understanding of existing algorithms

Abstract

In this work we propose a batch version of the Greenkhorn algorithm for multimarginal regularized optimal transport problems. Our framework is general enough to cover, as particular cases, some existing algorithms like Sinkhorn and Greenkhorn algorithm for the bi-marginal setting, and (greedy) MultiSinkhorn for multimarginal optimal transport. We provide a complete convergence analysis, which is based on the properties of the iterative Bregman projections (IBP) method with greedy control. Global linear rate of convergence and explicit bound on the iteration complexity are obtained. When specialized to above mentioned algorithms, our results give new insights and/or improve existing ones.