Regularizing Wasserstein distances has proved to be the key in the recent advances of optimal transport (OT) in machine learning. Most prominent is the entropic regularization of OT, which not only allows for fast computations and differentiation using Sinkhorn algorithm, but also improves stability with respect to data and accuracy in many numerical experiments. Theoretical understanding of these benefits remains unclear, although recent statistical works have shown that entropy-regularized OT mitigates classical OT's curse of dimensionality. In this paper, we adopt a more geometrical point of view, and show using Fenchel duality that any convex regularization of OT can be interpreted as ground cost adversarial. This incidentally gives access to a robust dissimilarity measure on the ground space, which can in turn be used in other applications. We propose algorithms to compute this robust cost, and illustrate the interest of this approach empirically.

本研究提出了一个新的正则化解释角度，即将正则化视为一种鲁棒性机制，展示了任何凸正则化的OT都可以被解释为接受对手--地面成本的方式。这同时可以在地面空间上提供鲁棒的不相似性度量方法，并提出了相应的算法和实验性说明了这种方法的优越性。

正则化最优输运的理论是基于地面成本对抗的