{
  "id": "2009.13040",
  "version": "v1",
  "published": "2020-09-28T03:23:54.000Z",
  "updated": "2020-09-28T03:23:54.000Z",
  "title": "Likelihood Landscape and Local Minima Structures of Gaussian Mixture Models",
  "authors": [
    "Yudong Chen",
    "Xumei Xi"
  ],
  "categories": [
    "stat.ML",
    "cs.LG",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "In this paper, we study the landscape of the population negative log-likelihood function of Gaussian Mixture Models with a general number of components. Due to nonconvexity, there exist multiple local minima that are not globally optimal, even when the mixture is well-separated. We show that all local minima share the same form of structure that partially identifies the component centers of the true mixture, in the sense that each local minimum involves a non-overlapping combination of fitting multiple Gaussians to a single true component and fitting a single Gaussian to multiple true components. Our results apply to the setting where the true mixture components satisfy a certain separation condition, and are valid even when the number of components is over-or under-specified. For Gaussian mixtures with three components, we obtain sharper results in terms of the scaling with the separation between the components.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-28T03:23:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gaussian mixture models",
      "local minimum",
      "local minima structures",
      "likelihood landscape",
      "true mixture components satisfy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}