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

arXiv: https://arxiv.org/abs/2609.35948 · HTML: https://mlsanity.com/paper/2609.35948/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 41 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 6 of 6)

- [Dense Associative Memory for Pattern Recognition](https://mlsanity.com/paper/1606.01164/) (2016)
- [On Sparse Modern Hopfield Model](https://mlsanity.com/paper/2309.12673/) (2023)
- [Position: Beyond Euclidean -- Foundation Models Should Embrace Non-Euclidean Geometries](https://mlsanity.com/paper/2504.08896/) (2025)
- [Dense Associative Memory with Epanechnikov Energy](https://mlsanity.com/paper/2506.10801/) (2025)
- [Text Has Curvature](https://mlsanity.com/paper/2602.13418/) (2026)
- [Towards Non-Euclidean Foundation Models: Advancing AI Beyond Euclidean Frameworks](https://mlsanity.com/paper/2505.14417/) (2025)

## Abstract (arXiv)

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 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.