{
  "abstract": "We study eigenpair estimation for the tensorized, density-weighted Laplace--Beltrami operator from $n$ independent data points sampled from a multi-manifold mixture distribution where manifolds may intersect. Let $d$ denote the largest intrinsic dimension of the manifold components and $s\\geq2$ the smoothness of the sampling densities. For simple, nontrivial eigenpairs, we establish the minimax rate $n^{-s/(d+2s)}$ for approximating the eigenvalue and eigenfunction of the tensorized Laplace--Beltrami operator, and show that a continuum plug-in estimator based on kernel density estimation attains the same rate when the supporting manifolds are known. We further analyze a manifold-agnostic graph Laplacian estimator and prove that its eigenpairs converge to those of the tensorized Laplace--Beltrami operator at rate $n^{-1/(2.5d-1)}$, up to logarithmic factors, improving upon previously known spectral convergence rates for multi-manifold learning. Numerical experiments on synthetic multi-manifold data corroborate the rates guaranteed by our theorems. These results characterize the information-theoretic difficulty of multi-manifold spectral estimation and quantify the additional cost of learning without oracle component information.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15892",
  "author_total": 2,
  "authors": [
    {
      "name": "Haoyu Chen"
    },
    {
      "name": "Chenghui Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/afwcgczgth2v7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15892",
  "fulltext_url": null,
  "id": "afwcgczgth2v7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15892",
  "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.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1703|Topological and Geometric Data Analysis"
    ]
  ],
  "openalex": "W7221764157",
  "pdf_url": "https://arxiv.org/pdf/2610.15892",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/afwcgczgth2v7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:26Z",
      "native_id": "2610.15892",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15892"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Minimax Rates and Improved Spectral Convergence for Multi-Manifold Clustering",
  "title_text": "Minimax Rates and Improved Spectral Convergence for Multi-Manifold Clustering",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:26Z",
  "url": "https://research.draconitic.com/paper/afwcgczgth2v7",
  "venue": "arXiv",
  "versions": []
}