Computing Kantorovich-Wasserstein Distances on $d$-dimensional histograms using $(d+1)$-partite graphs

TL;DR

This paper introduces a new method for efficiently computing the exact Kantorovich-Wasserstein distance between high-dimensional histograms by transforming the problem into a minimum cost flow on a specially constructed $(d+1)$-partite graph, demonstrating competitive performance on image and biomedical data.

Contribution

The paper develops a novel graph-based approach that reduces the computation of Kantorovich-Wasserstein distances to a minimum cost flow problem on a $(d+1)$-partite graph, enabling exact and efficient calculations.

Findings

Method is competitive with state-of-the-art algorithms.

Applicable to gray scale images and biomedical histograms.

Reduces high-dimensional optimal transport to a structured flow problem.

Abstract

This paper presents a novel method to compute the exact Kantorovich-Wasserstein distance between a pair of $d$-dimensional histograms having $n$ bins each. We prove that this problem is equivalent to an uncapacitated minimum cost flow problem on a $(d+1)$-partite graph with $(d+1)n$ nodes and $dn^{\frac{d+1}{d}}$ arcs, whenever the cost is separable along the principal $d$-dimensional directions. We show numerically the benefits of our approach by computing the Kantorovich-Wasserstein distance of order 2 among two sets of instances: gray scale images and $d$-dimensional biomedical histograms. On these types of instances, our approach is competitive with state-of-the-art optimal transport algorithms.