{
  "id": "2301.07825",
  "version": "v1",
  "published": "2023-01-19T00:01:45.000Z",
  "updated": "2023-01-19T00:01:45.000Z",
  "title": "XTrace: Making the most of every sample in stochastic trace estimation",
  "authors": [
    "Ethan N. Epperly",
    "Joel A. Tropp",
    "Robert J. Webber"
  ],
  "comment": "27 pages, 7 figures",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "The implicit trace estimation problem asks for an approximation of the trace of a square matrix, accessed via matrix-vector products (matvecs). This paper designs new randomized algorithms, XTrace and XNysTrace, for the trace estimation problem by exploiting both variance reduction and the exchangeability principle. For a fixed budget of matvecs, numerical experiments show that the new methods can achieve errors that are orders of magnitude smaller than existing algorithms, such as the Girard-Hutchinson estimator or the Hutch++ estimator. A theoretical analysis confirms the benefits by offering a precise description of the performance of these algorithms as a function of the spectrum of the input matrix. The paper also develops an exchangeable estimator, XDiag, for approximating the diagonal of a square matrix using matvecs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-19T00:01:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65C05",
      "65F30",
      "68W20"
    ],
    "keywords": [
      "stochastic trace estimation",
      "implicit trace estimation problem asks",
      "square matrix",
      "algorithms",
      "matrix-vector products"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}