Relaxation of optimal transport problem via strictly convex functions

TL;DR

This paper introduces a mathematical framework and an iterative gradient descent method for a relaxed optimal transport problem on finite spaces, utilizing strictly convex functions like the Kullback--Leibler divergence.

Contribution

It develops a new relaxation approach for optimal transport problems using Bregman divergences and provides an iterative solution method.

Findings

Provides a mathematical foundation for the relaxation approach.

Proposes an iterative gradient descent algorithm.

Enhances understanding of optimal transport in data sciences.

Abstract

An optimal transport problem on finite spaces is a linear program. Recently, a relaxation of the optimal transport problem via strictly convex functions, especially via the Kullback--Leibler divergence, sheds new light on data sciences. This paper provides the mathematical foundations and an iterative process based on a gradient descent for the relaxed optimal transport problem via Bregman divergences.