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This paper presents a novel method to compute the exactKantorovich-Wasserstein distance between a pair of $d$-dimensional histogramshaving $n$ bins each. We prove that this problem is equivalent to anuncapacitated 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 separablealong the principal $d$-dimensional directions. We show numerically thebenefits of our approach by computing the Kantorovich-Wasserstein distance oforder 2 among two sets of instances: gray scale images and $d$-dimensionalbiomedical histograms. On these types of instances, our approach is competitivewith state-of-the-art optimal transport algorithms.

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