On bounds for norms and conditioning of Wasserstein metric matrix

Abstract

For the Wasserstein-1 metric matrices of one- and two-dimensions, we prove the two guesses about their computational properties, which were proposed by Bai in 2024 (Linear Algebra Appl. 681(2024), 150-186). More specifically, for these matrices we prove their nonsingularity and symmetric positive definiteness, and derive sharper upper bounds on the norms of their inverses and on their condition numbers, under much more relaxed and realistic conditions imposed upon the involved problem and discretization parameters.