Sparsity of Quadratically Regularized Optimal Transport: Bounds on concentration and bias

TL;DR

This paper provides quantitative bounds on the support size and location of the optimal coupling in quadratically regularized optimal transport, revealing sparsity phenomena and rates depending on regularization and dimension.

Contribution

It offers the first quantitative analysis of support sparsity in QOT, including bounds on support size and location in general dimensions.

Findings

Support of optimal coupling is sparse for small regularization.

Derived bounds depend on regularization parameter and dimension.

In self-transport case, support size scales as psilon^{1/(2+d)}.

Abstract

We study the quadratically regularized optimal transport (QOT) problem for quadratic cost and compactly supported marginals $\mu$ and $\nu$. It has been empirically observed that the optimal coupling $\pi_\epsilon$ for the QOT problem has sparse support for small regularization parameter $\epsilon>0.$ In this article we provide the first quantitative description of this phenomenon in general dimension: we derive bounds on the size and on the location of the support of $\pi_\epsilon$ compared to the Monge coupling. Our analysis is based on pointwise bounds on the density of $\pi_\epsilon$ together with Minty's trick, which provides a quadratic detachment from the optimal transport duality gap. In the self-transport setting $\mu=\nu$ we obtain optimal rates of order $\epsilon^{\frac{1}{2+d}}.$