{
  "id": "1904.05352",
  "version": "v1",
  "published": "2019-04-10T19:33:51.000Z",
  "updated": "2019-04-10T19:33:51.000Z",
  "title": "Regularized divergences between covariance operators and Gaussian measures on Hilbert spaces",
  "authors": [
    "Minh Ha Quang"
  ],
  "comment": "52 pages",
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "This work presents an infinite-dimensional generalization of the correspondence between the Kullback-Leibler and R\\'enyi divergences between Gaussian measures on Euclidean space and the Alpha Log-Determinant divergences between symmetric, positive definite matrices. Specifically, we present the regularized Kullback-Leibler and R\\'enyi divergences between covariance operators and Gaussian measures on an infinite-dimensional Hilbert space, which are defined using the infinite-dimensional Alpha Log-Determinant divergences between positive definite trace class operators. We show that, as the regularization parameter approaches zero, the regularized Kullback-Leibler and R\\'enyi divergences between two equivalent Gaussian measures on a Hilbert space converge to the corresponding true divergences. The explicit formulas for the divergences involved are presented in the most general Gaussian setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-10T19:33:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28C20",
      "60G15",
      "47B65",
      "15A15"
    ],
    "keywords": [
      "gaussian measures",
      "hilbert space",
      "covariance operators",
      "regularized divergences",
      "renyi divergences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}