{
  "abstract": "Active subspaces identify low-dimensional linear structure in high-dimensional parameter-to-output maps by estimating the dominant eigenspace of a gradient covariance operator. In practice this covariance is replaced by a Monte Carlo estimator built from a limited number of gradient evaluations. Classical analyses based on controlling the covariance error in operator norm lead to sample-complexity estimates that can be substantially more pessimistic than the sampling rules commonly used in computations. This paper studies the empirical active subspace method directly in the projection-error metric relevant for ridge approximation. We derive non-asymptotic quasi-optimality bounds governed by a regularized inverse Christoffel function associated with the gradient field. Under a bounded-gradient assumption, the resulting estimates already improve the sample-complexity estimates obtained from operator-norm covariance bounds. We then show that additional smoothness of the gradient map, expressed through membership in a reproducing kernel Hilbert space, yields sharper coherence estimates and motivates tractable importance sampling from kernel diagonal measures. Furthermore, the same smoothness assumption yields a priori decay bounds for the population active subspace tail energy, which can be combined with our finite-sample estimate to prescribe rank, regularization scale, and sample size, allowing to fully characterize the a priori sample complexity. The abstract assumptions are verified for lognormal Gaussian and affine uniform parametric elliptic PDEs using weighted summability of Hermite and Legendre series expansions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08940",
  "author_total": 3,
  "authors": [
    {
      "name": "Fabio Nobile"
    },
    {
      "name": "Matteo Raviola"
    },
    {
      "name": "Raúl Tempone"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1y7ym53zbn0sq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08940",
  "fulltext_url": null,
  "id": "1y7ym53zbn0sq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08940",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7212050499",
  "pdf_url": "https://arxiv.org/pdf/2609.08940",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1y7ym53zbn0sq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.08940",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08940"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "On the sample complexity of the active subspace method",
  "title_text": "On the sample complexity of the active subspace method",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/1y7ym53zbn0sq",
  "venue": "arXiv",
  "versions": []
}