{
  "id": "cs/0703125",
  "version": "v1",
  "published": "2007-03-25T01:19:14.000Z",
  "updated": "2007-03-25T01:19:14.000Z",
  "title": "Intrinsic dimension of a dataset: what properties does one expect?",
  "authors": [
    "Vladimir Pestov"
  ],
  "comment": "6 pages, 6 figures, 1 table, latex with IEEE macros, final submission to Proceedings of the 22nd IJCNN (Orlando, FL, August 12-17, 2007)",
  "journal": "Proceedings of the 20th International Joint Conference on Neural Networks (IJCNN'2007), Orlando, Florida (Aug. 12--17, 2007), pp. 1775--1780.",
  "categories": [
    "cs.LG"
  ],
  "abstract": "We propose an axiomatic approach to the concept of an intrinsic dimension of a dataset, based on a viewpoint of geometry of high-dimensional structures. Our first axiom postulates that high values of dimension be indicative of the presence of the curse of dimensionality (in a certain precise mathematical sense). The second axiom requires the dimension to depend smoothly on a distance between datasets (so that the dimension of a dataset and that of an approximating principal manifold would be close to each other). The third axiom is a normalization condition: the dimension of the Euclidean $n$-sphere $\\s^n$ is $\\Theta(n)$. We give an example of a dimension function satisfying our axioms, even though it is in general computationally unfeasible, and discuss a computationally cheap function satisfying most but not all of our axioms (the ``intrinsic dimensionality'' of Ch\\'avez et al.)",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-03-25T01:19:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "properties",
      "first axiom postulates",
      "intrinsic dimensionality",
      "axiomatic approach",
      "high values"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007cs........3125P"
    }
  }
}