# Quadratically-Regularized Optimal Transport on Graphs

## TL;DR

This paper introduces a quadratic regularization method for optimal transport on graphs, providing theoretical insights and an efficient Newton-type algorithm, as an alternative to entropic regularization.

## Contribution

It proposes a novel quadratic regularization approach for graph optimal transport, with theoretical analysis and a practical second-order optimization algorithm.

## Key findings

- Quadratic regularization influences flow structure on graphs.
- Derived an efficient Newton-type optimization method.
- Analyzed behavior of flows under small regularization.

## Abstract

Optimal transportation provides a means of lifting distances between points on a geometric domain to distances between signals over the domain, expressed as probability distributions. On a graph, transportation problems can be used to express challenging tasks involving matching supply to demand with minimal shipment expense; in discrete language, these become minimum-cost network flow problems. Regularization typically is needed to ensure uniqueness for the linear ground distance case and to improve optimization convergence; state-of-the-art techniques employ entropic regularization on the transportation matrix. In this paper, we explore a quadratic alternative to entropic regularization for transport over a graph. We theoretically analyze the behavior of quadratically-regularized graph transport, characterizing how regularization affects the structure of flows in the regime of small but nonzero regularization. We further exploit elegant second-order structure in the dual of this problem to derive an easily-implemented Newton-type optimization algorithm.