Geometric Rate–Distortion Invariance for Domain Generalization

摘要

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.