{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35948",
  "author_total": 2,
  "authors": [
    {
      "name": "Krishnakumar Balasubramanian"
    },
    {
      "name": "Zhaoyang Shi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ennaj9jpvay72.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35948",
  "fulltext_url": null,
  "id": "ennaj9jpvay72",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35948",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.AI"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7214936592",
  "pdf_url": "https://arxiv.org/pdf/2609.35948",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ennaj9jpvay72.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:41Z",
      "native_id": "2609.35948",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35948"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Intrinsic Associative Memory on Riemannian Manifolds: Curvature, Capacity, and Emergent Modes",
  "title_text": "Intrinsic Associative Memory on Riemannian Manifolds: Curvature, Capacity, and Emergent Modes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:41Z",
  "url": "https://research.draconitic.com/paper/ennaj9jpvay72",
  "venue": "arXiv",
  "versions": []
}