Optimal Transport to a Variety

TL;DR

This paper investigates how to minimize Wasserstein distance between probability distributions and algebraic varieties, providing polynomial equation solutions and detailed analysis for specific models, with implications for optimal transport problems.

Contribution

It introduces a method to compute Wasserstein distance to algebraic varieties using polynomial systems, with detailed case analysis based on ground metrics.

Findings

Solution characterized by polynomial equations

Explicit analysis for two bit independence model

Dependence on ground metric for solution structure

Abstract

We study the problem of minimizing the Wasserstein distance between a probability distribution and an algebraic variety. We consider the setting of finite state spaces and describe the solution depending on the choice of the ground metric and the given distribution. The Wasserstein distance between the distribution and the variety is the minimum of a linear functional over a union of transportation polytopes. We obtain a description in terms of the solutions of a finite number of systems of polynomial equations. The case analysis is based on the ground metric. A detailed analysis is given for the two bit independence model.