Optimal transport problems regularized by generic convex functions: A geometric and algorithmic approach

TL;DR

This paper generalizes entropy-regularized optimal transport to include arbitrary convex functions, proposing an efficient iterative algorithm especially effective for quadratic regularization.

Contribution

It extends the regularization framework in optimal transport to generic convex functions and introduces a tailored iterative solution method.

Findings

The method efficiently solves regularized optimal transport problems with various convex functions.

Quadratic regularization yields particularly efficient solutions.

The approach broadens the applicability of optimal transport algorithms.

Abstract

In order to circumvent the difficulties in solving numerically the discrete optimal transport problem, in which one minimizes the linear target function $P\mapsto\langle C,P\rangle:=\sum_{i,j}C_{ij}P_{ij}$, Cuturi introduced a variant of the problem in which the target function is altered by a convex one $\Phi(P)=\langle C,P\rangle-\lambda\mathcal{H}(P)$, where $\mathcal{H}$ is the Shannon entropy and $\lambda$ is a positive constant. We herein generalize their formulation to a target function of the form $\Phi(P)=\langle C,P\rangle+\lambda f(P)$, where $f$ is a generic strictly convex smooth function. We also propose an iterative method for finding a numerical solution, and clarify that the proposed method is particularly efficient when $f(P)=\frac{1}{2}\|P\|^2$.