Optimal Transport and Wasserstein Barycenter for Radially Contoured Distributions

Abstract

The optimal transport and Wasserstein barycenter of Gaussian distributions have been solved. In literature, the closed form formulas of the Monge map, the Wasserstein distance and the Wasserstein barycenter have been given. Moreover, when Gaussian distributions extend more generally to elliptically contoured distributions, similar results also hold true. In this case, Gaussian distributions are regarded as elliptically contoured distribution with generator function $e^{-x/2}$. However, there are few results about optimal transport for elliptically contoured distributions with different generator functions. In this paper, we degenerate elliptically contoured distributions to radially contoured distributions and study their optimal transport and prove their Wasserstein barycenter is still radially contoured. For general elliptically contoured distributions, we give two numerical counterexamples to show that the Wasserstein barycenter of elliptically contoured distributions does not have to be elliptically contoured.