{
  "id": "2109.10659",
  "version": "v1",
  "published": "2021-09-22T11:31:45.000Z",
  "updated": "2021-09-22T11:31:45.000Z",
  "title": "Improved variants of the Hutch++ algorithm for trace estimation",
  "authors": [
    "David Persson",
    "Alice Cortinovis",
    "Daniel Kressner"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "This paper is concerned with two improved variants of the Hutch++ algorithm for estimating the trace of a square matrix, implicitly given through matrix-vector products. Hutch++ combines randomized low-rank approximation in a first phase with stochastic trace estimation in a second phase. In turn, Hutch++ only requires $O\\left(\\varepsilon^{-1}\\right)$ matrix-vector products to approximate the trace within a relative error $\\varepsilon$ with high probability. This compares favorably with the $O\\left(\\varepsilon^{-2}\\right)$ matrix-vector products needed when using stochastic trace estimation alone. In Hutch++, the number of matrix-vector products is fixed a priori and distributed in a prescribed fashion among the two phases. In this work, we derive an adaptive variant of Hutch++, which outputs an estimate of the trace that is within some prescribed error tolerance with a controllable failure probability, while splitting the matrix-vector products in a near-optimal way among the two phases. For the special case of symmetric positive semi-definite matrix, we present another variant of Hutch++, called Nystr\\\"om++, which utilizes the so called Nystr\\\"om approximation and requires only one pass over the matrix, as compared to two passes with Hutch++. We extend the analysis of Hutch++ to Nystr\\\"om++. Numerical experiments demonstrate the effectiveness of our two new algorithms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-22T11:31:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65C05",
      "65F30",
      "68W20",
      "68W25",
      "68W27",
      "68W40"
    ],
    "keywords": [
      "matrix-vector products",
      "stochastic trace estimation",
      "symmetric positive semi-definite matrix",
      "square matrix",
      "randomized low-rank approximation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}