{
  "abstract": "Cluster labels are often used twice: first to discover groups, and then to test whether the discovered groups differ. This double use of the data makes ordinary two-sample $p$-values anti-conservative. Selective inference fixes this by conditioning on the clustering event, but for spectral clustering the event appears difficult because the algorithm combines an affinity map, an eigendecomposition, and a rounding step. We show that, for spectral clustering based on unweighted $\\varepsilon$- or $k$-nearest-neighbor graphs, the obstruction is the affinity construction rather than the eigendecomposition. Along the one-dimensional perturbation used by the truncated-$χ$ selective test, graph changes occur only at finitely many real roots of data-dependent quadratic distance comparisons. Between these breakpoints, the graph, Laplacian spectrum, embedding, rounding, and reported cluster pair are deterministic functions of a finite graph state. Consequently, in real arithmetic and under a known-scale spherical Gaussian model, the selection set is an exact finite union of intervals, yielding finite-sample selective $p$-values for selected spectral-cluster pairs. The same argument gives feature-wise truncated-normal tests. We also record direct extensions to known feature covariance, eigengap-selected $K$, and auxiliary-coordinate graphs. We validate null and partial-null calibration, operational power, numerical sensitivity, and model-scope diagnostics, and give a working-model PBMC algorithmic illustration. Smooth kernels such as RBF affinities fall outside the finite-state guarantee.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04995",
  "author_total": 1,
  "authors": [
    {
      "name": "Naotoshi Nakamura"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/119b1fd34mkw9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04995",
  "fulltext_url": null,
  "id": "119b1fd34mkw9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04995",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04995",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/119b1fd34mkw9.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:34Z",
      "native_id": "2611.04995",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04995"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Exact Selective Inference for Spectral Clustering via Piecewise-Constant Graph Perturbation",
  "title_text": "Exact Selective Inference for Spectral Clustering via Piecewise-Constant Graph Perturbation",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:10Z",
  "url": "https://research.draconitic.com/paper/119b1fd34mkw9",
  "venue": "arXiv",
  "versions": []
}