Invariant Feature Extraction Through Conditional Independence and the Optimal Transport Barycenter Problem: the Gaussian case

This paper introduces a method for extracting invariant features that predict a response variable while mitigating the influence of confounding variables. The core idea involves penalizing statistical dependence between the extracted features and confounders, conditioned on the response variable. The authors cleverly replace this with a more practical independence condition using the Optimal Transport Barycenter Problem. A key result is the equivalence of these two conditions in the Gaussian case. Furthermore, the paper addresses the scenario where true confounders are unknown, suggesting the use of surrogate variables. The method provides a closed-form solution for linear feature extraction in the Gaussian case, and the authors claim it can be extended to non-Gaussian and non-linear scenarios. The reliance on Gaussian assumptions is a potential limitation.