Kantorovich distance on a finite metric space

Luigi Montrucchio; Giovanni Pistone

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arXiv · 1905.07547

Kantorovich distance on a finite metric space

Luigi Montrucchio·Giovanni Pistone

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Abstract

Kantorovich distance (or 1-Wasserstein distance) on the probability simplex of a finite metric space is the value of a Linear Programming problem for which a closed-form expression is known in some cases. When the ground distance is defined by a graph, a few examples have already been studied. In the present paper, after re-deriving, with different tools, the result for trees, we prove that, for an arbitrary weighted graph, the K-distance is the minimum of the K-distances over all the spanning trees associated with the graph. We work in the dual LP-problem by using Arens-Eells norm associated with the metric space. Finally, we introduce new norms that are naturally related to $\ell_1$-embeddable distances and allows for a partial extension of our results to this new setting.

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Luigi Montrucchio, Giovanni Pistone. 2019-05-18. Kantorovich distance on a finite metric space. https://arxiv.org/abs/1905.07547

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