Sparsity and uniform regularity for regularised optimal transport

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-$C^{1,α}$-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.