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).

翻译：机器学习中的许多问题可以表述为在概率测度空间上求解熵正则化最优传输。经典方法涉及以丰富数学性质著称的Sinkhorn迭代。近期，Sinkhorn算法被重新纳入镜像下降框架，从而受益于经典优化理论见解。在此，我们基于这一成果引入Sinkhorn算法的连续时间类比。该视角使我们能够推导出对噪声和偏差具有鲁棒性的Sinkhorn方案新变体。此外，我们的连续时间动力学不仅泛化了机器学习与数学中的若干最新动力学（如（Deb等人，2023）的“Wasserstein镜像流”或（Claisse等人，2023）的“均场薛定谔方程”），还提供了统一视角。