E-ROBOT: a dimension-free method for robust statistics and machine learning via Schr"odinger bridge

We propose the Entropic-regularized Robust Optimal Transport (E-ROBOT) framework, a novel method that combines the robustness of ROBOT with the computational and statistical benefits of entropic regularization. We show that, rooted in the Schr"odinger bridge problem theory, E-ROBOT defines the robust Sinkhorn divergence overlineWvarepsilon,λ, where the parameter λ controls robustness and varepsilon governs the regularization strength. Letting n∈ mathbbN denote the sample size, a central theoretical contribution is establishing that the sample complexity of overlineWvarepsilon,λ is O(n-1/2), thereby avoiding the curse of dimensionality that plagues standard ROBOT. This dimension-free property unlocks the use of overlineWvarepsilon,λ as a loss function in large-dimensional statistical and machine learning tasks. With this regard, we demonstrate its utility through four applications: goodness-of-fit testing; computation of barycenters for corrupted 2D and 3D shapes; definition of gradient flows; and image colour transfer. From the computation standpoint, a perk of our novel method is that it can be easily implemented by modifying existing (textttPython) routines. From the theoretical standpoint, our work opens the door to many research directions in statistics and machine learning: we discuss some of them.