Scaling Algorithms for Unbalanced Transport Problems

TL;DR

This paper develops fast, parallelizable algorithms based on entropic regularization to efficiently solve unbalanced optimal transport problems, extending classical methods like Sinkhorn to handle arbitrary positive measures.

Contribution

It introduces a generalized entropic regularization scheme for unbalanced optimal transport, enabling efficient computation of transport, gradient flows, and barycenters for unbalanced measures.

Findings

Algorithms are highly parallelizable and involve simple diagonal scaling.

Applications include shape modification, color transfer, and growth modeling.

Methods outperform traditional approaches in speed and flexibility.

Abstract

This article introduces a new class of fast algorithms to approximate variational problems involving unbalanced optimal transport. While classical optimal transport considers only normalized probability distributions, it is important for many applications to be able to compute some sort of relaxed transportation between arbitrary positive measures. A generic class of such "unbalanced" optimal transport problems has been recently proposed by several authors. In this paper, we show how to extend the, now classical, entropic regularization scheme to these unbalanced problems. This gives rise to fast, highly parallelizable algorithms that operate by performing only diagonal scaling (i.e. pointwise multiplications) of the transportation couplings. They are generalizations of the celebrated Sinkhorn algorithm. We show how these methods can be used to solve unbalanced transport, unbalanced gradient flows, and to compute unbalanced barycenters. We showcase applications to 2-D shape modification, color transfer, and growth models.