{
  "abstract": "Multivariate angular data lie on a torus, but clustering methods often use the flat torus. We introduce curved-torus energy (CTE), a position-dependent dissimilarity based on restricted Riemannian path energy and minimization over winding classes. CTE has a certified finite winding reduction, an exact flat endpoint, a geodesic upper bound, and stability with respect to the geometry parameter. For label-free geometry selection, we propose CEHS-HM, a held-out criterion for selecting the warping parameter at a specified clustering resolution. We establish a finite-grid population-score regret bound and also consider joint geometry-resolution selection over finite candidate grids. Controlled experiments on $\\mathbb{T}^3$ show that CTE agrees with the flat method at zero warping and becomes more useful as geometric warping increases. Additional experiments show that the CEHS-selected geometry should be viewed as a data-adaptive clustering choice rather than as an estimator of a generative deformation parameter. Under a product-von-Mises mechanism, the selector tends toward the flat endpoint when curvature is unnecessary. On ILE protein torsion data, the proposed method identifies a stable five-mode representation of joint backbone-side-chain conformation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24079",
  "author_total": 3,
  "authors": [
    {
      "name": "Surojit Biswas"
    },
    {
      "name": "Soumyadip Paik"
    },
    {
      "name": "Buddhananda Banerjee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/768f3zxj5yt5a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24079",
  "fulltext_url": null,
  "id": "768f3zxj5yt5a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24079",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "openalex",
      "1702|Advanced Clustering Algorithms Research"
    ]
  ],
  "openalex": "W7223039964",
  "pdf_url": "https://arxiv.org/pdf/2610.24079",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/768f3zxj5yt5a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:33:54Z",
      "native_id": "2610.24079",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24079"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ME",
      "subfield": "statistics"
    }
  ],
  "title": "Geometry-Adaptive Clustering on High-Dimensional Tori",
  "title_text": "Geometry-Adaptive Clustering on High-Dimensional Tori",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:54Z",
  "url": "https://research.draconitic.com/paper/768f3zxj5yt5a",
  "venue": "arXiv",
  "versions": []
}