Optimal Transport with Tempered Exponential Measures

TL;DR

This paper introduces a new approach to optimal transport using tempered exponential measures, balancing computational efficiency and plan sparsity, and extending to unbalanced transport scenarios.

Contribution

It extends entropic-regularized optimal transport to tempered exponential measures, achieving a balance between sparsity and computational speed, and naturally incorporating unbalanced transport.

Findings

Fast approximation algorithms for tempered exponential measures.

Controlled sparsity in optimal transport plans.

Applicability to unbalanced optimal transport problems.

Abstract

In the field of optimal transport, two prominent subfields face each other: (i) unregularized optimal transport, "\`a-la-Kantorovich", which leads to extremely sparse plans but with algorithms that scale poorly, and (ii) entropic-regularized optimal transport, "\`a-la-Sinkhorn-Cuturi", which gets near-linear approximation algorithms but leads to maximally un-sparse plans. In this paper, we show that an extension of the latter to tempered exponential measures, a generalization of exponential families with indirect measure normalization, gets to a very convenient middle ground, with both very fast approximation algorithms and sparsity, which is under control up to sparsity patterns. In addition, our formulation fits naturally in the unbalanced optimal transport problem setting.