Learning on Hyperbolic Domains with Geodesic Multiscale Filters

Hyperbolic learning usually treats feature vectors or latent states as points in a hyperbolic representation space. We study a complementary regime in which hyperbolicity belongs to the domain: observations remain Euclidean-valued signals attached to points of a negatively curved space. We construct intrinsic multiscale operators from differences of radial geodesic Gaussians and use their cascades to define a hyperbolic scattering representation. Because the kernels depend only on geodesic distance, the representation transforms equivariantly under global hyperbolic isometries. We then introduce HScat, which retains scattering-path responses and learns task-adaptive channel mixing without requiring manifold-valued feature propagation. On six node-classification benchmarks, HScat attains the highest mean F1 among the evaluated Euclidean, hyperbolic, and geometry-aware methods. These early results support a signal-processing view of hyperbolic learning and motivate broader study of multiscale operators on negatively curved domains.