{
  "id": "2103.00815",
  "version": "v1",
  "published": "2021-03-01T07:30:31.000Z",
  "updated": "2021-03-01T07:30:31.000Z",
  "title": "Computation complexity of deep ReLU neural networks in high-dimensional approximation",
  "authors": [
    "Dinh Dũng",
    "Van Kien Nguyen",
    "Mai Xuan Thao"
  ],
  "comment": "30 pages. arXiv admin note: text overlap with arXiv:2007.08729",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "The purpose of the present paper is to study the computation complexity of deep ReLU neural networks to approximate functions in H\\\"older-Nikol'skii spaces of mixed smoothness $H_\\infty^\\alpha(\\mathbb{I}^d)$ on the unit cube $\\mathbb{I}^d:=[0,1]^d$. In this context, for any function $f\\in H_\\infty^\\alpha(\\mathbb{I}^d)$, we explicitly construct nonadaptive and adaptive deep ReLU neural networks having an output that approximates $f$ with a prescribed accuracy $\\varepsilon$, and prove dimension-dependent bounds for the computation complexity of this approximation, characterized by the size and the depth of this deep ReLU neural network, explicitly in $d$ and $\\varepsilon$. Our results show the advantage of the adaptive method of approximation by deep ReLU neural networks over nonadaptive one.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-01T07:30:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "computation complexity",
      "high-dimensional approximation",
      "adaptive deep relu neural networks",
      "dimension-dependent bounds",
      "unit cube"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}