{
  "id": "1206.4868",
  "version": "v1",
  "published": "2012-06-21T13:24:55.000Z",
  "updated": "2012-06-21T13:24:55.000Z",
  "title": "An inequality for relative entropy and logarithmic Sobolev inequalities in Euclidean spaces",
  "authors": [
    "Katalin Marton"
  ],
  "comment": "29 pages (in PDF format) Submitted to Journal of Functional Analysis. This paper tackles the same problem as arXiv:0907.4491, but gives a more satisfactory result, and makes arXiv:0907.4491 superfluous",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $q(x)$ and $p(x)$ denote density functions on the $n$-dimensional Euclidean space, and let $p_i(\\cdot|y_1,..., y_{i-1},y_{i+1},..., y_n)$ and $Q_i(\\cdot|x_1,..., x_{i-1},x_{i+1},..., x_n)$ denote their local specifications. For a class of density functions $q$ we prove an inequality between the relative entropy $D(p||q)$ and a weighted sum of the conditional relative entropies $D(p_i(\\cdot|Y_1,..., Y_{i-1},Y_{i+1},..., Y_n) ||Q_i(\\cdot|Y_1,..., Y_{i-1},Y_{i+1},..., Y_n))$ that holds for any $p$. The weights are proportional to the logarithmic Sobolev constants of the local specifications of $q$. Thereby we derive a logarithmic Sobolev inequality for a weighted Gibbs sampler governed by the local specifications of $q$. Moreover, this inequality implies a classical logarithmic Sobolev inequality for $q$, as defined for Gaussian distribution by L. Gross. This strengthens a result by F. Otto and M. Reznikoff. The proof is based on ideas developed by F. Otto and C. Villani in their paper on the connection between Talagrand's transportation-cost inequality and logarithmic Sobolev inequality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-06-21T13:24:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A40",
      "60K35",
      "82B20",
      "82C22"
    ],
    "keywords": [
      "relative entropy",
      "local specifications",
      "talagrands transportation-cost inequality",
      "classical logarithmic sobolev inequality",
      "logarithmic sobolev constants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.4868M"
    }
  }
}