We show how to break the curse of dimension for the estimation of optimal transport distance between two smooth distributions for the Euclidean squared distance. The approach relies on essentially one tool: represent inequality constraints in the dual formulation of OT by equality constraints with a sum of squares in reproducing kernel Hilbert space. By showing this representation is tight in the variational formulation, one can then leverage smoothness to break the curse.
Oct 21, 2021 2:00 PM
Université d'Avignon, département de mathématiques.
After presenting some background on optimal transport, we present entropic regularization, its link with the Schr”odinger problem and the Sinkhorn algorithm. We present unbalanced optimal transport and the corresponding Sinkhorn algorithm. Then, we show how to improve on the vanilla Sinkhorn algorithm in this particular case. Then, we switch to a different problem which is the statistical estimation of optimal transport. Under smoothness assumptions on the transport maps we achieve a parametric rate of estimation of the distance using a sum-of-squares in Sobolev spaces.
I explain the formulation of the Wasserstein-Fisher-Rao distance, introduced a few years ago as the natural extension of the Wasserstein L2 metric to the space of positive Radon measures. We present the equivalence between the dynamic formulation and a static formulation, where we relax the marginal constraints with relative entropy. Then, we will present a connection with standard optimal transport on a cone space. The second part of the talk is motivated by this optimal transport distance and we study a generalized Camassa-Holm equation, for which we study existence of minimizing generalized geodesics, a la Brenier.