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

arXiv:2609.35948v1 Announce Type: new 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 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.

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 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.

Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Machine Learning (stat.ML)

arXiv:2609.35948 [cs.LG]

(or arXiv:2609.35948v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.35948 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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

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