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

Part of Advances in Neural Information Processing Systems 31 (NeurIPS 2018)

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Gennaro Auricchio, Federico Bassetti, Stefano Gualandi, Marco Veneroni


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.