Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration

TL;DR

This paper proves that optimal transport distances can be approximated in near-linear time using Sinkhorn iteration, introduces a new greedy algorithm called Greenkhorn, and demonstrates its practical efficiency.

Contribution

It provides a theoretical analysis showing near-linear time approximation for optimal transport and introduces Greenkhorn, a more efficient greedy coordinate descent algorithm.

Findings

Greenkhorn outperforms Sinkhorn in practice

Sinkhorn iteration achieves near-linear time approximation

New analysis of Sinkhorn iteration underpins the algorithms

Abstract

Computing optimal transport distances such as the earth mover's distance is a fundamental problem in machine learning, statistics, and computer vision. Despite the recent introduction of several algorithms with good empirical performance, it is unknown whether general optimal transport distances can be approximated in near-linear time. This paper demonstrates that this ambitious goal is in fact achieved by Cuturi's Sinkhorn Distances. This result relies on a new analysis of Sinkhorn iteration, which also directly suggests a new greedy coordinate descent algorithm, Greenkhorn, with the same theoretical guarantees. Numerical simulations illustrate that Greenkhorn significantly outperforms the classical Sinkhorn algorithm in practice.