Regularity of Solutions to Beckmann's Parametric Optimal Transport

TL;DR

This paper develops a regularity theory for solutions to Beckmann's optimal transport problem, deriving H"older regularity of solutions and their dependence on parameters, enabling neural network approximation.

Contribution

It introduces a novel regularity analysis for Beckmann's problem using elliptic PDE techniques and extends the results to parameter-dependent distributions and other probability flows.

Findings

H"older regularity of potential, flux, and flow solutions derived

Conditions for H"older continuity of solutions with respect to parameters established

Neural network approximation of solutions in H"older norm demonstrated

Abstract

Beckmann's problem in optimal transport minimizes the total squared flux in a continuous transport problem from a source to a target distribution. In this article, the regularity theory for solutions to Beckmann's problem in optimal transport is developed utilizing an unconstrained Lagrangian formulation and solving the variational first order optimality conditions. It turns out that the Lagrangian multiplier that enforces Beckmann's divergence constraint fulfills a Poisson equation and the flux vector field is obtained as the potential's gradient. Utilizing Schauder estimates from elliptic regularity theory, the exact H\"older regularity of the potential, the flux and the flow generating is derived on the basis of H\"older regularity of source and target densities on a bounded, regular domain. If the target distribution depends on parameters, as is the case in conditional (``promptable'') generative learning, we provide sufficient conditions for separate and joint H\"older continuity of the resulting vector field in the parameter and the data dimension. Following a recent result by Belomnestny et al., one can thus approximate such vector fields with deep ReQu neural networks in C^(k,alpha)-H\"older norm. We also show that this approach generalizes to other probability paths, like Fisher-Rao gradient flows.