Intrinsic Associative Memory on Riemannian Manifolds: Curvature, Capacity, and Emergent Modes

תקציר מקורי באנגליתarXiv:2609.35948v1 Announce Type: cross Abstract: Geometry does more than constrain an associative memory: curvature determines what it remembers and which states it creates. We develop intrinsic dense associative memories on Riemannian manifolds by casting memory as Epanechnikov kernel-density mode seeking. We compare geodesic and volume-corrected energies and show that curvature separates their behavior. We prove that geodesic memory always retains an isolated pattern, while corrected memory obeys a sharp Ricci-curvature threshold: positive curvature can erase memories in high dimensions, while negative curvature reinforces them. We derive geodesic capacity scalings of $q_\beta^{-1/2}$ for retaining every pattern and $q_\beta^{-1}$ for a typical one, where $q_\beta$ is the pairwise kerne