Wasserstein Mirror Gradient Flow as the limit of the Sinkhorn Algorithm

TL;DR

This paper establishes that the Sinkhorn algorithm's marginals converge to a Wasserstein mirror gradient flow as regularization vanishes, linking optimal transport, PDEs, and stochastic processes.

Contribution

It introduces the concept of Sinkhorn flow as a Wasserstein mirror gradient flow and connects it to PDEs and stochastic diffusions, providing new theoretical insights.

Findings

Convergence of Sinkhorn marginals to the Wasserstein mirror gradient flow.

Identification of the flow as a solution to a parabolic Monge-Ampère PDE.

Conditions for exponential convergence of the Sinkhorn flow.

Abstract

We prove that the sequence of marginals obtained from the iterations of the Sinkhorn algorithm or the iterative proportional fitting procedure (IPFP) on joint densities, converges to an absolutely continuous curve on the $2$-Wasserstein space, as the regularization parameter $\varepsilon$ goes to zero and the number of iterations is scaled as $1/\varepsilon$ (and other technical assumptions). This limit, which we call the Sinkhorn flow, is an example of a Wasserstein mirror gradient flow, a concept we introduce here 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. An equivalent description of this flow is provided by the parabolic Monge-Amp\`{e}re PDE whose connection to the Sinkhorn algorithm was noticed by Berman (2020). We derive conditions for exponential convergence for this limiting flow. We also construct a Mckean-Vlasov diffusion whose marginal distributions follow the Sinkhorn flow.