Tropical optimal transport and Wasserstein distances

Abstract

We study the problem of optimal transport in tropical geometry and define the Wasserstein-p distances in the continuous metric measure space setting of the tropical projective torus. We specify the tropical metric—a combinatorial metric that has been used to study of the tropical geometric space of phylogenetic trees—as the ground metric and study the cases of $$p=1,2$$ p = 1 , 2 in detail. The case of $$p=1$$ p = 1 gives an efficient computation of the infinitely-many geodesics on the tropical projective torus, while the case of $$p=2$$ p = 2 gives a form for Fréchet means and a general inner product structure. Our results also provide theoretical foundations for geometric insight a statistical framework in a tropical geometric setting. 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 in tropical geometry. Several numerical examples are provided.

When optimal transport meets information geometry

“…In the past five years, a related line of work was advanced by Li and his colleagues, [15,[41][42][43][44][45], under the name transport information geometry. Specifically, this work compares the Fisher-Rao metric with its dual connections and the 2-Wasserstein metric "side-by-side."…”

Tropical Geometric Variation of Tree Shapes

“…This equivalence relation generates a quotient space known as the tropical projective torus, denoted by R n /R1, which is the ambient space of tree space, U N ⊂ T N ⊂ R n /R1. The tropical projective torus R n /R1 may be embedded into R n−1 by considering representatives of the equivalence classes with first coordinate equal to zero, [24,28,29] for more detail.…”

“…This indicates a more complex geometry than the CAT(0) structure of BHV space. Lee et al [24] use optimal transport theory to define and study the Wasserstein distances on the tropical projective torus and thus give an algorithm to compute the set of all infinitely-many geodesics on the tropical projective torus.…”