Intrinsic Associative Memory on Riemannian Manifolds: Curvature, Capacity, and Emergent Modes
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 qβ−1/2 for retaining every pattern and qβ−1 for a typical one, where qβ is the pairwise kernel-overlap probability. We show how overlap \emph{creates} novel memories: designed N-pattern configurations realize all 2N−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.
Related
- Infinitesimal Hilbertianity for manifolds with Sobolev regular Riemannian metrics through uniform lower capacity bounds
- Local Topology of Riemannian Manifolds with Lower Intermediate Ricci Curvature Bounds
- Hopf type theorem for surfaces of constant weighted mean curvature in Riemannian manifolds
- Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvature
- Intrinsic Rigidity of Bar-and-Joint Frameworks on Riemannian Manifolds
BibTeX
@misc{balasubramanian2026intrinsic,
title = {{Intrinsic Associative Memory on Riemannian Manifolds: Curvature, Capacity, and Emergent Modes}},
author = {Krishnakumar Balasubramanian and Zhaoyang Shi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35948},
eprint = {2609.35948},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35948},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Intrinsic Associative Memory on Riemannian Manifolds: Curvature, Capacity, and Emergent Modes
AU - Krishnakumar Balasubramanian
AU - Zhaoyang Shi
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35948
UR - https://arxiv.org/abs/2609.35948
AB - 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.
ER -