A fast algorithm for Earth Mover's Distance based on optimal transport and L1 type Regularization

TL;DR

This paper introduces a fast algorithm for approximating Earth Mover's Distance by reformulating it as an L1 minimization problem with regularization, leveraging primal-dual methods from compressed sensing for rapid convergence.

Contribution

The paper presents a novel regularized L1 minimization approach for EMD, enabling faster computation using primal-dual algorithms inspired by compressed sensing techniques.

Findings

The proposed algorithm converges rapidly in numerical experiments.

It provides accurate EMD approximations efficiently.

The method benefits from simple iterative updates.

Abstract

We propose a new algorithm to approximate the Earth Mover's distance (EMD). Our main idea is motivated by the theory of optimal transport, in which EMD can be reformulated as a familiar $L_1$ type minimization. We use a regularization which gives us a unique solution for this $L_1$ type problem. The new regularized minimization is very similar to problems which have been solved in the fields of compressed sensing and image processing, where several fast methods are available. In this paper, we adopt a primal-dual algorithm designed there, which uses very simple updates at each iteration and is shown to converge very rapidly. Several numerical examples are provided.