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

2026-09-28

AI Summary

This study addresses how the curvature of Riemannian manifolds governs the capacity, pattern preservation, and novel state generation of associative memory. The proposed approach models memory as kernel density mode seeking, employing Epanechnikov kernel estimation and a Riemannian mean shift algorithm. By contrasting geodesic and volume-corrected energies, it theoretically establishes a Ricci curvature threshold effect and derives capacity scaling laws. This work reveals that pattern overlap can generate emergent states, positioning curvature as a fundamental design variable for memory systems. Simulations validate the predicted curvature transition phenomena, while experiments on WordNet hierarchical retrieval demonstrate significant performance improvements in low-capacity regimes.

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 $q_β^{-1/2}$ for retaining every pattern and $q_β^{-1}$ for a typical one, where $q_β$ is the pairwise kernel-overlap probability. We show how overlap \emph{creates} novel memories: designed $N$-pattern configurations realize all $2^N-1$ subset modes, but random data at the storage threshold yield only a Poisson number. We establish exact one-step recall using Riemannian mean shift. In simulations, we recover the predicted curvature transition and every designed mode. On WordNet's full noun hierarchy, we demonstrate that volume correction improves low-capacity retrieval. Together, our work shows that curvature is a design variable for associative memory, not merely a property of the data.

Problem

• Associative Memory
• Riemannian Manifolds
• Curvature
• Memory Capacity
• Emergent Modes

Innovation

• Riemannian manifolds
• Dense associative memory
• Ricci curvature
• Kernel density mode seeking
• Riemannian mean shift