Work

Sparsity of Quadratically Regularized Optimal Transport: Bounds on Concentration and Bias

Year: 2025

Type: article

Abstract: . We study the quadratically regularized optimal transport (QOT) problem for quadratic cost and compactly supported marginals [Formula: see text] and [Formula: see text]. It has been empirically observed that the optimal coupling [Formula: see text] for the QOT problem has sparse support for small regularization parameter [Formula: see text]. In this article, we provide the first quantitative description of this phenomenon in the general dimension: we derive bounds on the size and the location of the support of [Formula: see text] compared to those of the Monge coupling. Our analysis is based on pointwise bounds on the density of [Formula: see text] together with Minty’s trick, which provides a quadratic detachment from the optimal transport duality gap. In the self-transport setting [Formula: see text], we obtain optimal rates of order [Formula: see text].

Source: SIAM Journal on Mathematical Analysis

DOI: 10.1137/25m1723633

Authors: Johannes Wiesel, Xingyu Xu

Institutions: University of Copenhagen, Carnegie Mellon University

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, Division of Electrical, Communications and Cyber Systems

Awards: Taming Nonlinear Inverse Problems: Theory and Algorithms, G6691530897, Optimal Transport of Stochastic Processes in Mathematical Finance