B1905
Title: Wasserstein mirror gradient flows as the limit of the Sinkhorn algorithm
Authors: Nabarun Deb - University of Chicago (United States) [presenting]
Abstract: The sequence of marginals obtained from iterations of the Sinkhorn or IPFP algorithm is studied and it is shown that under a suitable time and regularization scaling, the marginals converge to an absolutely continuous curve on the Wasserstein space. The limit, which we call the Sinkhorn flow, is an example of a Wasserstein mirror gradient flow, a concept introduced which is inspired by the well-known Euclidean mirror gradient flows. In the case of Sinkhorn, the gradient is that of the relative entropy functional with respect to one of the marginals and the mirror is half of the squared Wasserstein distance functional from the other marginal. Interestingly, the norm of the velocity field of this flow can be interpreted as the metric derivative with respect to the linearized optimal transport (LOT) distance. Examples are provided to show that these flows can have faster convergence rates than usual gradient flows. A Mckean Vlasov SDE is also constructed whose marginal distributions give rise to the same flow.
