{
  "abstract": "We study initialization bias in spherical $K$-means for weakly informative directional mixtures. We model the observations by a $K$-component von Mises-Fisher mixture with a small concentration parameter $κ$, corresponding to a high-dispersion regime in which the data provide limited information about the underlying directions. Our analysis begins with the limiting case $κ=0$ (corresponding to a uniform distribution over the sphere), where one population spherical $K$-means update is governed entirely by the Voronoi tessellation induced by the initialized templates. For uniformly random initializations in fixed dimension $d$, the updated templates become asymptotically aligned with their initial values as $K\\to\\infty$: the average squared geodesic error scales as $O(K^{-2/(d-1)})$, while the worst-case error is $O((\\log K/K)^{2/(d-1)})$. We then show that, in the weak-signal regime of small positive $κ$, the population update remains an $O(κ)$ perturbation of this limiting map. Thus, in the weak-signal regime, spherical $K$-means can preserve initialization-induced structure despite the presence of a genuine but highly dispersed directional signal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02205",
  "author_total": 2,
  "authors": [
    {
      "name": "Amnon Balanov"
    },
    {
      "name": "Tamir Bendory"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/43zq0h5x26wfq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02205",
  "fulltext_url": null,
  "id": "43zq0h5x26wfq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02205",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "eess.SP"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Bayesian Methods and Mixture Models"
    ]
  ],
  "openalex": "W7207809414",
  "pdf_url": "https://arxiv.org/pdf/2609.02205",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/43zq0h5x26wfq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02205",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02205"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "A Geometric Analysis of Initialization Bias in Spherical $K$-means in the Weak Signal Regime",
  "title_text": "A Geometric Analysis of Initialization Bias in Spherical K-means in the Weak Signal Regime",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/43zq0h5x26wfq",
  "venue": "arXiv",
  "versions": []
}