{
  "id": "1312.3580",
  "version": "v1",
  "published": "2013-12-12T18:37:03.000Z",
  "updated": "2013-12-12T18:37:03.000Z",
  "title": "Bounding the smallest singular value of a random matrix without concentration",
  "authors": [
    "Vladimir Koltchinskii",
    "Shahar Mendelson"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Given $X$ a random vector in ${\\mathbb{R}}^n$, set $X_1,...,X_N$ to be independent copies of $X$ and let $\\Gamma=\\frac{1}{\\sqrt{N}}\\sum_{i=1}^N <X_i,\\cdot>e_i$ be the matrix whose rows are $\\frac{X_1}{\\sqrt{N}},\\dots, \\frac{X_N}{\\sqrt{N}}$. We obtain sharp probabilistic lower bounds on the smallest singular value $\\lambda_{\\min}(\\Gamma)$ in a rather general situation, and in particular, under the assumption that $X$ is an isotropic random vector for which $\\sup_{t\\in S^{n-1}}{\\mathbb{E}}|<t,X>|^{2+\\eta} \\leq L$ for some $L,\\eta>0$. Our results imply that a Bai-Yin type lower bound holds for $\\eta>2$, and, up to a log-factor, for $\\eta=2$ as well. The bounds hold without any additional assumptions on the Euclidean norm $\\|X\\|_{\\ell_2^n}$. Moreover, we establish a nontrivial lower bound even without any higher moment assumptions (corresponding to the case $\\eta=0$), if the linear forms satisfy a weak `small ball' property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-12T18:37:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "smallest singular value",
      "random matrix",
      "bai-yin type lower bound holds",
      "concentration",
      "sharp probabilistic lower bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.3580K"
    }
  }
}