A Statistical Learning Perspective on Semi-dual Adversarial Neural Optimal Transport Solvers

TL;DR

This paper provides a theoretical analysis of adversarial neural optimal transport methods, establishing generalization error bounds for semi-dual quadratic OT solvers using neural networks, and discusses potential extensions to general OT problems.

Contribution

It introduces the first statistical learning framework for analyzing adversarial neural OT solvers, deriving bounds based on neural network properties, and suggests future work on general OT cases.

Findings

Derived upper bounds on generalization error for semi-dual quadratic OT neural networks.

Bounds depend on standard properties of neural network function classes.

Experimental results support the theoretical findings.

Abstract

Neural network-based optimal transport (OT) is a recent and fruitful direction in the generative modeling community. It finds its applications in various fields such as domain translation, image super-resolution, computational biology and others. Among the existing OT approaches, of considerable interest are adversarial minimax solvers based on semi-dual formulations of OT problems. While promising, these methods lack theoretical investigation from a statistical learning perspective. Our work fills this gap by establishing upper bounds on the generalization error of an approximate OT map recovered by the minimax quadratic OT solver. Importantly, the bounds we derive depend solely on some standard statistical and mathematical properties of the considered functional classes (neural nets). While our analysis focuses on the quadratic OT, we believe that similar bounds could be derived for general OT case, paving the promising direction for future research. Our experimental illustrations are available online https://github.com/milenagazdieva/StatOT.