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

05/18/2018

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by   Gennaro Auricchio, et al.

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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^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.