2024/10/04 by Alberto González-Sanz, Marcel Nutz, González-Sanz, Alberto +1 · 4 citations
Computer Science · Mathematics · #49N05 #49N10 #90C25 #Advanced Mathematical Modeling in Engineering #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2410.03353
openalex publication_date 2024/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The quadratically regularized optimal transport problem is empirically known to have sparse solutions: its optimal coupling πε has sparse support for small regularization parameter ε, in contrast to entropic regularization whose solutions have full support for any ε>0. Focusing on continuous and scalar marginals, we provide the first precise description of this sparsity. Namely, we show that the support of πε shrinks to the Monge graph at the sharp rate ε1/3. This result is based on a detailed analysis of the dual potential fε for small ε. In particular, we prove that fε is twice differentiable a.s. and bound the second derivative uniformly in ε, showing that fε is uniformly strongly convex. Convergence rates for fε and its derivative are also obtained.