Geometric Rate–Distortion Invariance for Domain Generalization

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 Rate–Distortion Invariance (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.