An Algorithm to Approximate the Optimal Expected Inner Product of Two Vectors with Given Marginals

Abstract

We introduce a new algorithm, called the swapping algorithm, to approximate numerically the minimal and maximal expected inner product of two random vectors with given marginal distributions. As a direct application, the algorithm computes an approximation of the L2-Wasserstein distance between two multivariate measures. The algorithm is simple to implement, accurate and less computationally expensive than the algorithms generally used in the literature for this problem. The algorithm also provides a discretized image of optimal measures and can be extended to more general cost functionals.

Keywords: Swapping algorithm, Covariance between vectors, p-Wasserstein distance, Earth Mover’s Distance, Linear Sum Assignment Problem, Optimal transportations.

JEL Classification: C61