Rate of convergence of support in L^p-regularized optimal transport

Determine the asymptotic rate, as the regularization parameter ε→0, at which the support of the L^p-regularized optimal transport plan π_ε (for quadratic cost with p∈(1,2]) between probability measures λ and μ converges to the support of the unregularized optimal transport plan of the classical optimal transport problem.

Background

The paper studies L^p-regularized optimal transport (ROT) with quadratic cost and p∈(1,2], where the ROT optimizer πε has density that can vanish, leading to sparsity. Under suitable conditions, the support of πε shrinks to the support of the classical optimal transport (OT) solution as ε→0. While this convergence was known qualitatively, the outstanding question concerns its quantitative rate.

The authors obtain sharp local rates away from the boundary by proving that, for interior points, the conditional support sections behave like balls with radius of order ε^{1/(d(p−1)+2)}, and derive corresponding strong convexity and convergence-rate results for the potentials. The abstract explicitly identifies the rate of support convergence as the main open question motivating these results.