The Wasserstein metric matrix and its computational property

Abstract

By further exploring and deeply analyzing the concrete algebraic structures and essential computational properties about the Wasserstein-1 metric matrices of one- and two-dimensions, we show that they can be essentially expressed by the Neumann series of nilpotent matrices and, therefore, the products of these matrices with a prescribed vector can be accomplished accurately and stably in the optimal computational complexities through solving unit bidiagonal triangular systems of linear equations. We also give appropriate generalizations of these one- and two-dimensional Wasserstein-1 metric matrices, as well as their corresponding extensions to higher dimensions, and demonstrate the algebraic structures and computational properties of these generalized and extended Wasserstein-1 metric matrices.