Sum-of-Squares Programming for Ma-Trudinger-Wang Regularity of Optimal Transport Maps

TL;DR

This paper introduces a computational Sum-of-Squares approach to verify the Ma-Trudinger-Wang tensor's non-negativity, expanding the class of cost functions with guaranteed regularity in optimal transport maps.

Contribution

It develops a provably correct SOS programming method to certify MTW tensor non-negativity and approximate regions of regularity for various ground costs.

Findings

SOS method successfully certifies MTW non-negativity for multiple cost functions.

The approach provides inner approximations of regularity regions.

Application to practical costs demonstrates effectiveness in real scenarios.

Abstract

For a given ground cost, approximating the Monge optimal transport map that pushes forward a given probability measure onto another has become a staple in several modern machine learning algorithms. The fourth-order Ma-Trudinger-Wang (MTW) tensor associated with this ground cost function provides a notion of curvature in optimal transport. The non-negativity of this tensor plays a crucial role for establishing continuity for the Monge optimal transport map. It is, however, generally difficult to analytically verify this condition for any given ground cost. To expand the class of cost functions for which MTW non-negativity can be verified, we propose a provably correct computational approach which provides certificates of non-negativity for the MTW tensor using Sum-of-Squares (SOS) programming. We further show that our SOS technique can also be used to compute an inner approximation of the region where MTW non-negativity holds. We apply our proposed SOS programming method to several practical ground cost functions to approximate the regions of regularity of their corresponding optimal transport maps.