{
  "id": "2106.13345",
  "version": "v1",
  "published": "2021-06-24T22:58:51.000Z",
  "updated": "2021-06-24T22:58:51.000Z",
  "title": "The Hanson-Wright Inequality for Random Tensors",
  "authors": [
    "Stefan Bamberger",
    "Felix Krahmer",
    "Rachel Ward"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We provide moment bounds for expressions of the type $(X^{(1)} \\otimes \\dots \\otimes X^{(d)})^T A (X^{(1)} \\otimes \\dots \\otimes X^{(d)})$ where $\\otimes$ denotes the Kronecker product and $X^{(1)}, \\dots, X^{(d)}$ are random vectors with independent, mean 0, variance 1, subgaussian entries. The bounds are tight up to constants depending on $d$ for the case of Gaussian random vectors. Our proof also provides a decoupling inequality for expressions of this type. Using these bounds, we obtain new, improved concentration inequalities for expressions of the form $\\|B (X^{(1)} \\otimes \\dots \\otimes X^{(d)})\\|_2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-24T22:58:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random tensors",
      "hanson-wright inequality",
      "expressions",
      "gaussian random vectors",
      "moment bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}