🤖 AI Summary
This work investigates the continuous limit behavior of Sinkhorn iterations (i.e., the iterative proportional fitting procedure) in the 2-Wasserstein space as the regularization parameter ε → 0 and the number of iterations scales as 1/ε. Methodologically, we introduce the novel concept of Wasserstein mirror gradient flow, integrating relative entropy gradient analysis, asymptotic expansion of Monge–Ampère-type PDEs, and McKean–Vlasov diffusion construction. We rigorously establish existence and uniqueness of the limiting curve, proving it is absolutely continuous in Wasserstein space; its velocity field admits an explicit characterization via metric derivatives and is exactly reproduced by a first-order stochastic differential equation. This yields the first rigorous connection between Sinkhorn iteration and parabolic optimal transport dynamics, leading to an exponential convergence criterion and revealing Sinkhorn’s intrinsic nature as an implicit gradient flow on the Wasserstein manifold.
📝 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.
