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.

翻译：我们证明，当正则化参数 $\varepsilon$ 趋近于零且迭代次数按 $1/\varepsilon$ 缩放时（以及其他技术假设），由联合密度上的Sinkhorn算法或迭代比例拟合过程（IPFP）迭代得到的边际分布序列，在 $2$-Wasserstein空间上收敛于一条绝对连续曲线。该极限——我们称之为Sinkhorn流——是Wasserstein镜像梯度流的一个实例，这一概念受著名的欧几里得镜像梯度流启发而引入。在Sinkhorn情形下，梯度是相对于其中一个边际的相对熵泛函的梯度，而镜像则源自另一边际的二分之一平方Wasserstein距离泛函。有趣的是，该流速度场的范数可解释为关于线性化最优传输（LOT）距离的度量导数。该流的一个等价描述由抛物型Monge-Ampère偏微分方程给出，Berman（2020）曾注意到该方程与Sinkhorn算法的关联。我们推导了该极限流指数收敛的条件，并构造了一个其边际分布遵循Sinkhorn流的McKean-Vlasov扩散过程。