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 ignore discriminative structure can become ill-conditioned under finite samples. This calls for shaping the geometry of class-conditional representations across domains, not merely matching their distributions. We propose Geometric R ate– D istortion I nvariance ( RDI ), a DG framework that realizes this principle by generalizing classical rate–distortion theory to Grassmann manifolds. RDI 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), promoting stable alignment while preventing the collapse of discriminative geometry. We provide finite-sample stability guarantees under bounded shifts and show on DomainBed that RDI is competitive with strong DG baselines, with ablations confirming that both alignment and complexity control are necessary for reliable generalization.