Wasserstein distance to an algebraic variety

A metric d on the finite set {1,…,n} induces a metric on the (n-1)-dimensional probability simplex which is called Wasserstein distance. This distance is well known to statisticians, and possesses properties which make it appealing to several areas of computer science. Moreover, its unit ball is a polytope, whose combinatorial structure encodes information on the metric d. We study the problem of computing the Wasserstein distance from a point to a statistical model described by polynomial equations. After providing all the necessary definitions, I will discuss the combinatorical and algebraic complexity of this problem, with a special focus on models of independent random variables. This is joint work with Türkü Özlüm Çelik, Asgar Jamneshan, Guido Montúfar and Bernd Sturmfels.