Cumulant Perturbations of Gaussian Optimal Transport and Wasserstein-2 Metric

Abstract

In this thesis we extend the closed form solution for quadratic cost optimal transport on Gaussian densities to densities perturbed from Gaussians in the 3rd order cumulant, as asymptotically approximated by an Edgeworth series. This is done through a linearization in our perturbative factor ϵ of the Monge-Ampère equation for smooth quadratic cost optimal transport. This results in an Ornstein-Uhlenbeck PDE, which we solve to give us the transport map T(x) = m1 + A(x−m0)−ϵ(A:1 ∆κ(3)) : H2(x−m0; Σ0). From this map we compute the Wasserstein-2 metric between such densities and the displacement interpolation formula for the 3rd cumulant.