General construction and classes of explicit L1-optimal couplings

Abstract

The main scope of this paper is to give some explicit classes of examples of $L^1$-optimal couplings. Optimal transportation w.r.t. the Kantorovich metric ${\ell }_1$ (resp. the Wasserstein metric $W_1$) between two absolutely continuous measures is known since the basic papers of Kantorovich and Rubinstein (Dokl. Akad. Nauk SSSR 115 (1957) 1058–1061) and Sudakov (Proc. Steklov Inst. Math. 141 (1979) 1–178) to occur on rays induced by a decomposition of the basic space (and more generally to higher dimensional decompositions in the case of general measures) 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 few particular instances.

First, we give a self-contained review of some basic optimal coupling results and we propose and investigate in particular some basic principles for the construction of $L^1$-optimal couplings given by a reduction principle and some usable forms of the decomposition method. This reduction principle, together with symmetry properties of the reduced measures, gives a hint to the decomposition of the space into sectors and via the non crossing property of optimal transport leads to the choice of transportation rays. The optimality of the induced transports is then a consequence of the characterization results of optimal couplings.

Then, we apply these principles to determine in explicit form $L^1$-optimal couplings for several classes of examples of elliptical distributions. In particular, we give for the first time a general construction of $L^1$-optimal couplings between two bivariate Gaussian distributions. We also discuss optimality of special constructions like shifts and scalings, and provide an extended class of dual functionals allowing for the closed-form computation of the ${\ell }_1$-metric or of accurate lower bounds of it in a variety of examples.