General Construction and Classes of Explicit L 1-optimal Couplings
37 Pages Posted: 5 Jun 2021
Date Written: May 28, 2021
Optimal transportation w.r.t. the Kantorovich metric l1 (resp. the Wasser- stein metric W1) between two absolutely continuous measures is known since the basic papers of Kantorovich and Rubinstein (1957) and Sudakov (1979) to occur on rays induced by a decomposition of the basic space, which is induced by the corresponding dual potentials. Several papers have given this kind of structural result and established existence and uniqueness of solutions in varying generality. Since the dual problems pose typically too strong challenges to be solved in explicit form these structural results have so far been applied for the solution of very few particular instances. In this paper we propose and investigate as basic principles for the construction of L1-optimal couplings a reduction principle and some usable forms of the decomposition method. As our main contribution we apply these principles to determine in explicit form L1-optimal couplings for several classes of examples of elliptical distributions. In particular, we give for the first time the general construction of L1-optimal couplings between two bivariate Gaussian distributions. We also discuss optimality of special constructions like shifts and scaling, and provide an extended class of dual functionals allowing for the closed-form computation of the l1-metric or of accurate lower bounds of it in a variety of examples.
Keywords: Kantorovich l1-Metric, L1-Wasserstein Distance, Optimal Mass Transportation, Optimal Couplings, Gaussian Distributions, Monge-Kantorovich Problem, Kantorovich-Rubinstein Theorem
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