Sparsity and uniform regularity for regularised optimal transport

TL;DR

This paper establishes uniform regularity estimates for regularised quadratic optimal transport problems, demonstrating convergence of regularised solutions to unregularised ones and providing sharp bounds on the support size, applicable to various regularisation types.

Contribution

It provides new uniform regularity estimates and support bounds for regularised optimal transport, extending previous results to subquadratic and entropic regularisations.

Findings

Interior Lipschitz estimates for transport maps and potentials.

Convergence of regularised solutions to unregularised ones in specific function spaces.

Sharp local support bounds for regularised optimal transport plans.

Abstract

We consider regularised quadratic optimal transport with subquadratic polynomial or entropic regularisation. In both cases, we prove interior Lipschitz-estimates on a transport-like map and interior gradient Lipschitz-estimates on the potentials, under the assumption that the transport map solving the unregularised problem is bi-$C^{1,\alpha}$-regular. For strictly subquadratic and entropic regularisation, the estimates improve to interior $C^1$ and $C^2$ estimates for the transport-like map and the potentials, respectively. Our estimates are uniform in the regularisation parameter. As a consequence of this, we obtain convergence of the transport-like map (resp. the potentials) to the unregularised transport map (resp. Kantorovich potentials) in $C^{0,1-}_{\mathrm{loc}}$ (resp. $C^{1,1-}_{\mathrm{loc}}$). Central to our approach are sharp local bounds on the size of the support for regularised optimal transport which we derive for a general convex, superlinear regularisation term. These bounds are of independent interest and imply global bias bounds for the regularised transport plans. Our global bounds, while not necessarily sharp, improve on the best known results in the literature for quadratic regularisation.