{
  "id": "2101.07564",
  "version": "v1",
  "published": "2021-01-19T11:18:51.000Z",
  "updated": "2021-01-19T11:18:51.000Z",
  "title": "Performance analysis of greedy algorithms for minimising a Maximum Mean Discrepancy",
  "authors": [
    "Luc Pronzato"
  ],
  "comment": "34 pages, 7 figures, preprint submitted to a journal",
  "categories": [
    "stat.ML",
    "cs.LG",
    "stat.CO"
  ],
  "abstract": "We analyse the performance of several iterative algorithms for the quantisation of a probability measure $\\mu$, based on the minimisation of a Maximum Mean Discrepancy (MMD). Our analysis includes kernel herding, greedy MMD minimisation and Sequential Bayesian Quadrature (SBQ). We show that the finite-sample-size approximation error, measured by the MMD, decreases as $1/n$ for SBQ and also for kernel herding and greedy MMD minimisation when using a suitable step-size sequence. The upper bound on the approximation error is slightly better for SBQ, but the other methods are significantly faster, with a computational cost that increases only linearly with the number of points selected. This is illustrated by two numerical examples, with the target measure $\\mu$ being uniform (a space-filling design application) and with $\\mu$ a Gaussian mixture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-19T11:18:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "62K99",
      "65D30",
      "65D99"
    ],
    "keywords": [
      "maximum mean discrepancy",
      "greedy algorithms",
      "performance analysis",
      "greedy mmd minimisation",
      "approximation error"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}