Stability of the Monge Map in Semi-Dual Optimal Transport

TL;DR

This paper analyzes the stability of Monge maps in semi-dual optimal transport, revealing a saddle-point structure and conditions for convergence that explain practical algorithm behavior.

Contribution

It provides a theoretical framework for understanding the convergence of Monge maps in semi-dual optimal transport without assuming dual optimality.

Findings

Semi-dual formulation has a degenerate saddle-point structure

Necessary and sufficient conditions for Monge map convergence are derived

Explains why numerical algorithms need more iterations in practice

Abstract

This paper shows that the semi-dual formulation of the optimal transport problem has a degenerate saddle-point structure, and that its numerical solution is equivalent to solving a constrained optimization problem. We derive necessary and sufficient conditions for the convergence of Monge maps without requiring optimality of the dual potential. This analysis helps explain why, in practice, numerical algorithms often require more iterations to update the transport map than the potential.