Geometric Rate–Distortion Invariance for Domain Generalization

Abstract

Domain generalization (DG) aims to learn representations that remain predictive under distribution shifts. A key challenge is that the target domain is unobserved during training, which complicates the search for invariant representations: alignment objectives that do not account for the preservation of discriminative structure may become ill-conditioned or lead to degenerate solutions, especially under finite samples. We propose Geometric **R**ate–**D**istortion **I**nvariance (**RDI**), a DG framework that addresses this challenge by generalizing classical rate–distortion theory to Grassmann manifolds. **RDI** explicitly models class-conditional representations as low-dimensional subspaces and formulates DG as a joint optimization of (i) cross-domain subspace alignment (geometric distortion) and (ii) spectral–volumetric complexity (a capacity-regularized rate term). This integrated approach is designed to promote stable alignment while preventing the collapse of discriminative geometry, adapting to dataset-specific regimes. We provide finite-sample stability guarantees under bounded shifts. Experiments on DomainBed demonstrate that **RDI** is competitive with strong DG baselines, and ablations verify that reliable generalization necessitates the concerted action of both alignment and complexity control.