Stability of the Monge Map in Semi-Dual Optimal Transport

2026-05-06

AI Summary

This study investigates the convergence and stability of Monge maps in semi-dual optimal transport. Motivated by the empirical observation that map updates in numerical algorithms converge significantly slower than dual potentials, the authors analyze the degenerate saddle-point structure of the semi-dual formulation and reformulate it equivalently as a constrained optimization problem. Without requiring the dual potentials to attain optimality, they establish, for the first time, necessary and sufficient conditions for the convergence of Monge maps, thereby uncovering the intrinsic mechanism underlying the slow map iteration. This work provides a rigorous theoretical foundation for understanding and improving existing numerical algorithms in this domain.

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.

Problem

• Monge map
• optimal transport
• semi-dual formulation
• stability
• convergence

Innovation

• semi-dual optimal transport
• Monge map stability
• degenerate saddle-point
• constrained optimization
• dual potential convergence