# Tropical Optimal Transport and Wasserstein Distances

arXiv: https://arxiv.org/abs/1911.05401 · HTML: https://mlsanity.com/paper/1911.05401/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Nov 2019.

## Built on by (most cited first, 1 of 1)

- [OPAL: Orthonormal Prototype Alignment Learning for Interpretable Image Classification](https://mlsanity.com/paper/2608.30003/) (2026)

## Abstract (arXiv)

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 in detail. The case of p=1 gives an efficient computation of the infinitely-many geodesics on the tropical projective torus, while the case of p=2 gives a form for Fr'echet 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.