Ordinary differential equations for regularized variational problems involving semi-discrete optimal transport

TL;DR

This paper introduces a novel ODE-based approach to analyze and compute solutions for regularized variational problems involving semi-discrete optimal transport, providing a continuous parameter perspective.

Contribution

It characterizes solutions via well-posed ODEs in the regularization parameter, enabling robust numerical solutions without initial guess dependencies.

Findings

Solutions can be described by ODEs in the regularization parameter.

The solution trajectory is continuous, allowing recovery of unregularized solutions.

The ODE approach outperforms Newton's method in robustness.

Abstract

We consider entropically regularized, semi-discrete versions of variational problems on the set of probability measures involving optimal transport as well as other terms. We prove that the solutions can be characterized by well-posed ordinary differential equations in the regularization parameter. The initial conditions for these equations, corresponding to solutions to completely regularized problems, are typically known explicitly. The ODE can then be solved to recover the solution for an arbitrary degree of regularization; we verify that the solution is continuous in the regularization parameter, implying that taking the limit of the trajectory yields the solution to the fully unregularized problem. We establish analogous results for a version of the problem when the non-optimal transport term is not scaled with the regularization parameter. We exploit our characterization to numerically solve several example problems using standard ODE methods; this strategy exhibits superior robustness to alternatives such as Newton's method, as arbitrary initializations are not required.