The Wasserstein distance has been proved to be a valuable instrument in many applied fields where being able to evaluate the difference between pictures is of key importance, like image retrieval, color transfer, and machine learning. As the interest for this distance grew, the research for fast ways to compute this distance for discrete measures grew accordingly, since, even if it can be formulated as a simple linear problem, the computation of the W2 distance requires a huge number of unknowns. Under suitable assumptions on the cost function, we are able to reformulate the minimum problem associated to the W2 distance between two discrete measures as a linear programming problem with a reduced number of variables, which can be efficiently solved with a Network Simplex algorithm. Finally, we show how the same approach can be extended when the measures are not discrete. This allows us to prove a Pythagorean theorem for the W2 distance. The talk is based on joint work with S. Gualandi, M. Veneroni and F. Bassetti.
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