{
  "abstract": "Partial Least Squares (PLS) regression extracts a few outcome-aligned directions in a high-dimensional X and is widely used across applied science, but inference on the resulting fit is either expensive, biased and discouraged, or absent. We reduce inference to held-out OLS refits of the supervised subspace, a primitive shared by PLS, supervised PCA, and linear probes, and supply two tests using held-out correlations: a Nadeau-Bengio corrected asymptotic t-test as a fast approximation, and a permutation test with comparable power, finite-sample valid under outcome-predictor independence and iid rows. Held-out predictions are unchanged under any orthogonal rebasing of the supervised span, so an interpretable basis such as varimax inherits the joint claim but not a per-axis p-value; per-component claims come from a fixed-sequence test on the PLS extraction order. We validate on synthetic geometries, two NIR chemometric datasets, and cross-lingual word-embedding regressions; the exact test also transfers to supervised PCA and a ridge probe. The proposed tests have more power than CV-permutation-Q^2, at a fraction of its cost. A pre-run check on n and the spectrum of X says when the approximation is safe. We release a Rust library with Python, R, and Julia bindings, plus a Python text pipeline.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36307",
  "author_total": 2,
  "authors": [
    {
      "name": "Paweł Lenartowicz"
    },
    {
      "name": "Hubert Plisiecki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e8xhc59bmw42h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36307",
  "fulltext_url": null,
  "id": "e8xhc59bmw42h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36307",
  "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",
      "2613|Statistical Methods and Applications"
    ]
  ],
  "openalex": "W7214987320",
  "pdf_url": "https://arxiv.org/pdf/2609.36307",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e8xhc59bmw42h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:41Z",
      "native_id": "2609.36307",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36307"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Cheap and Powerful Tests for Supervised Subspaces: Per-Component Inference for PLS",
  "title_text": "Cheap and Powerful Tests for Supervised Subspaces: Per-Component Inference for PLS",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:41Z",
  "url": "https://research.draconitic.com/paper/e8xhc59bmw42h",
  "venue": "arXiv",
  "versions": []
}