{
  "abstract": "Implicit trace estimation aims to approximate the trace of a matrix or linear operator accessible only through matrix-vector or operator-vector products. In the matrix setting, the Girard-Hutchinson estimator typically requires $\\mathcal{O}(\\varepsilon^{-2})$ products to achieve accuracy $\\varepsilon$, while the variance-reduced Hutch++ algorithm reduces this sample complexity to $\\mathcal{O}(\\varepsilon^{-1})$ for positive semidefinite matrices. We develop infinite-dimensional analogues of these estimators for positive trace-class operators on separable Hilbert spaces. The idealized estimators use Gaussian random elements whose covariance is determined by the target operator, leading to unbiased operator versions of Girard-Hutchinson and Hutch++. We prove high-probability error bounds analogous to the finite-dimensional matrix results; in particular, idealized infHutch++ achieves $\\mathcal{O}(\\varepsilon^{-1})$ sample complexity. For practical computation, we introduce truncated implementations that restrict the random samples to finite-dimensional subspaces; for fixed sample budget, we show that truncated infHutch++ converges in distribution to its idealized counterpart as the truncation dimension tends to infinity. Numerical experiments with integral operators, density-of-states approximations, and spectral filtering for a radial Dirac operator show that these truncated estimators can achieve accuracy comparable to the ContHutch++ algorithm by Zvonek, Horning & Townsend while using lower-degree function representations and smaller internal discretizations in chebfun.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28135",
  "author_total": 4,
  "authors": [
    {
      "name": "Zvonimir Bujanović"
    },
    {
      "name": "Luka Grubišić"
    },
    {
      "name": "Daniel Kressner"
    },
    {
      "name": "Hrvoje Olić"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9ea37jxkgfgyt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28135",
  "fulltext_url": null,
  "id": "9ea37jxkgfgyt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28135",
  "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.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7204750198",
  "pdf_url": "https://arxiv.org/pdf/2608.28135",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9ea37jxkgfgyt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.28135",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28135"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Stochastic trace estimation for positive trace-class operators",
  "title_text": "Stochastic trace estimation for positive trace-class operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/9ea37jxkgfgyt",
  "venue": "arXiv",
  "versions": []
}