  Many problems in machine learning can be formulated as solving
entropy-regularized optimal transport on the space of probability measures. The
canonical approach involves the Sinkhorn iterates, renowned for their rich
mathematical properties. Recently, the Sinkhorn algorithm has been recast
within the mirror descent framework, thus benefiting from classical
optimization theory insights. Here, we build upon this result by introducing a
continuous-time analogue of the Sinkhorn algorithm. This perspective allows us
to derive novel variants of Sinkhorn schemes that are robust to noise and bias.
Moreover, our continuous-time dynamics not only generalize but also offer a
unified perspective on several recently discovered dynamics in machine learning
and mathematics, such as the "Wasserstein mirror flow" of (Deb et al. 2023) or
the "mean-field Schr\"odinger equation" of (Claisse et al. 2023).
