Tropical Optimal Transport and Wasserstein Distances in Phylogenetic Tree Space

Abstract

We study the problem of optimal transport on phylogenetic tree space from the perspective of tropical geometry, and thus define the Wasserstein-p distances for probability measures in this continuous metric measure space setting. With respect to the tropical metric—a combinatorial metric on the space of phylogenetic trees—the cases of $p=1,2$ are treated in detail, which give an efficient way to compute geodesics and also provide theoretical foundations for geometric insight a statistical framework on tree space. We construct explicit algorithms for the computation of the tropical Wasserstein-1 and 2 distances, and prove their convergence. Our results provide the first study of the Wasserstein distances and optimal transport on sets of phylogenetic trees. Several numerical examples are provided.

Wonjun Lee
Wonjun Lee
Graduate Student
Wuchen Li
Wuchen Li
Assistant Professor
Bo Lin
Bo Lin
Visiting Assistant Professor

My research interests include mathematical biology, tropical geometry and combinatorics.

Anthea Monod
Anthea Monod
Lecturer (Assistant Professor with tenure)

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