Sparsity of Quadratically Regularized Optimal Transport: Scalar Case

TL;DR

This paper precisely characterizes the sparsity of solutions in quadratically regularized optimal transport with scalar marginals, showing the support shrinks at a rate of ^{1/3} and analyzing the dual potential's properties.

Contribution

It provides the first detailed description of the sparsity pattern and support shrinking rate in quadratically regularized optimal transport for scalar cases.

Findings

Support shrinks to the Monge graph at ^{1/3} rate.

Dual potential is twice differentiable and uniformly strongly convex.

Convergence rates for the dual potential and its derivative are established.

Abstract

The quadratically regularized optimal transport problem is empirically known to have sparse solutions: its optimal coupling $\pi_{\varepsilon}$ has sparse support for small regularization parameter $\varepsilon$, in contrast to entropic regularization whose solutions have full support for any $\varepsilon>0$. Focusing on continuous and scalar marginals, we provide the first precise description of this sparsity. Namely, we show that the support of $\pi_{\varepsilon}$ shrinks to the Monge graph at the sharp rate $\varepsilon^{1/3}$. This result is based on a detailed analysis of the dual potential $f_{\varepsilon}$ for small $\varepsilon$. In particular, we prove that $f_{\varepsilon}$ is twice differentiable a.s. and bound the second derivative uniformly in $\varepsilon$, showing that $f_{\varepsilon}$ is uniformly strongly convex. Convergence rates for $f_{\varepsilon}$ and its derivative are also obtained.