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\"{o}dinger bridge problem theory, E-ROBOT defines the robust Sinkhorn divergence $\overline{W}_{\varepsilon,\lambda}$, where the parameter $\lambda$ controls robustness and $\varepsilon$ governs the regularization strength. Letting $n\in \mathbb{N}$ denote the sample size, a central theoretical contribution is establishing that the sample complexity of $\overline{W}_{\varepsilon,\lambda}$ is $\mathcal{O}(n^{-1/2})$, thereby avoiding the curse of dimensionality that plagues standard ROBOT. This dimension-free property unlocks the use of $\overline{W}_{\varepsilon,\lambda}$ 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 (\texttt{Python}) routines. From the theoretical standpoint, our work opens the door to many research directions in statistics and machine learning: we discuss some of them.