# Quadratically regularized optimal transport

## TL;DR

This paper explores quadratic regularization in optimal transport, deriving dual formulations, proving strong duality, and developing algorithms that produce sparse transport plans, contrasting with entropic regularization.

## Contribution

It introduces quadratic regularization for optimal transport, providing dual problem derivations, solution existence proofs, and two algorithms with numerical analysis.

## Key findings

- Algorithms perform well even with small regularization parameters
- Quadratic regularization yields sparse optimal transport plans
- Dual problem formulation and strong duality established

## Abstract

We investigate the problem of optimal transport in the so-called Kantorovich form, i.e. given two Radon measures on two compact sets, we seek an optimal transport plan which is another Radon measure on the product of the sets that has these two measures as marginals and minimizes a certain cost function. We consider quadratic regularization of the problem, which forces the optimal transport plan to be a square integrable function rather than a Radon measure. We derive the dual problem and show strong duality and existence of primal and dual solutions to the regularized problem. Then we derive two algorithms to solve the dual problem of the regularized problem: A Gauss-Seidel method and a semismooth quasi-Newton method and investigate both methods numerically. Our experiments show that the methods perform well even for small regularization parameters. Quadratic regularization is of interest since the resulting optimal transport plans are sparse, i.e. they have a small support (which is not the case for the often used entropic regularization where the optimal transport plan always has full measure).