2024/08/15 by Alberto González-Sanz, Marcel Nutz, González-Sanz, Alberto +3 · 2 citations
Computer Science · Engineering · Mathematics · #49N05 #49N10 #90C25 #Advanced Optimization Algorithms Research #Combinatorics (math.CO) #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Mathematical Programming #Optimization and Variational Analysis #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2408.07871
openalex publication_date 2024/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In optimal transport, quadratic regularization is a sparse alternative to entropic regularization: the solution measure tends to have small support. Computational experience suggests that the support decreases monotonically to the unregularized counterpart as the regularization parameter is relaxed. We find it useful to investigate this monotonicity more abstractly for linear programs over polytopes, regularized with the squared norm. Here, monotonicity can be stated as an invariance property of the curve mapping the regularization parameter to the solution: once the curve enters a face of the polytope, does it remain in that face forever? We show that this invariance is equivalent to a geometric property of the polytope, namely that each face contains the minimum norm point of its affine hull. Returning to the optimal transport problem and its associated Birkhoff polytope, we verify this property for low dimensions, but show that it fails for marginals with five or more point masses. As a consequence, the conjectured monotonicity of the support fails in general, even if experiments suggest that monotonicity holds for many cost matrices. Separately, we apply our geometric point of view to a problem of Erdős, namely to characterize the doubly stochastic matrices whose maximal trace equals their squared norm.