Barycenteric distribution alignment and manifold-restricted invertibility for domain generalization

We revisit the problem of Domain Generalization (DG) where the hypotheses are composed of a common representation mapping followed by a labeling function. Popular DG methods optimize a well-known upper bound to the risk in the unseen domain. However, the bound contains a term that is not optimized due to its dual dependence on the representation mapping and the unknown optimal labeling function for the unseen domain. We derive a new upper bound free of the term having such dual dependence by imposing mild assumptions on the loss function and an invertibility requirement on the representation map when restricted to the low-dimensional data manifold. The derivation leverages old and recent transport inequalities that link optimal transport metrics with information-theoretic measures. Our bound motivates a new algorithm for DG comprising Wasserstein-2 barycenter cost for feature alignment and mutual information or autoencoders for enforcing approximate invertibility. Experiments on several datasets demonstrate superior performance compared to well-known DG algorithms.