Infinitesimal behavior of Quadratically Regularized Optimal Transport and its relation with the Porous Medium Equation

Abstract: The quadratically regularized optimal transport problem has recently been considered in various applications where the coupling needs to be \emph{sparse}, i.e., the density of the coupling needs to be zero for a large subset of the product of the supports of the marginals. However, unlike the acclaimed entropy-regularized optimal transport, the effect of quadratic regularization on the transport problem is not well understood from a mathematical standpoint. In this work, we take a first step towards its understanding. We prove that the difference between the cost of optimal transport and its regularized version multiplied by the ratio $\varepsilon^{-\frac{2}{d+2}}$ converges to a nontrivial limit as the regularization parameter $\varepsilon$ tends to 0. The proof confirms a conjecture from Zhang et al. (2023) where it is claimed that a modification of the self-similar solution of the porous medium equation, the Barenblatt--Pattle solution, can be used as an approximate solution of the regularized transport cost for small values of $\varepsilon$.