On Stability Estimate of Optimal Transport Maps Using m-th Polynomial Convexity

Abstract

In 1781, Gaspard Monge first proposed the practical problem of relocating building materials while minimizing workers’ effort. Mathematically, the problem can be reiterated as finding a mapping T0 that transforms a random variable (X) following probability measure (μ) into a random variable (Y) following probability measure (ν), with minimal cost. Afterward, it has been widely studied and applied in statistics, machine learning, and economics, which concern the study of “distance” between usually a pair of probability distributions. The focus of this paper is centered on investigating and generalizing stability estimates for optimal transport plans, particularly through the lens of strong polynomial convexity. Building on previous research using plug-in estimators to strengthen the convergence rate of discrete or semi-discrete estimators for optimal transport plans, this paper introduces a novel stability estimate leveraging L-Lipschitz continuity and a paradigmatic methodology based on polynomial convexity, the understanding of which remains limited.