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We prove that the sequence of marginals obtained from the iterations of theSinkhorn algorithm or the iterative proportional fitting procedure (IPFP) onjoint densities, converges to an absolutely continuous curve on the$2$-Wasserstein space, as the regularization parameter $\varepsilon$ goes tozero and the number of iterations is scaled as $1/\varepsilon$ (and othertechnical assumptions). This limit, which we call the Sinkhorn flow, is anexample of a Wasserstein mirror gradient flow, a concept we introduce hereinspired by the well-known Euclidean mirror gradient flows. In the case ofSinkhorn, the gradient is that of the relative entropy functional with respectto one of the marginals and the mirror is half of the squared Wassersteindistance functional from the other marginal. Interestingly, the norm of thevelocity field of this flow can be interpreted as the metric derivative withrespect to the linearized optimal transport (LOT) distance. An equivalentdescription of this flow is provided by the parabolic Monge-Amp\`{e}re PDEwhose connection to the Sinkhorn algorithm was noticed by Berman (2020). Wederive conditions for exponential convergence for this limiting flow. We alsoconstruct a Mckean-Vlasov diffusion whose marginal distributions follow theSinkhorn flow.

Wasserstein Mirror Gradient Flow as the limit of the Sinkhorn Algorithm