{
  "id": "1910.14067",
  "version": "v1",
  "published": "2019-10-30T18:09:29.000Z",
  "updated": "2019-10-30T18:09:29.000Z",
  "title": "Spectral properties of kernel matrices in the flat limit",
  "authors": [
    "Simon Barthelmé",
    "Konstantin Usevich"
  ],
  "comment": "40 pages, 8 pages",
  "categories": [
    "math.NA",
    "cs.NA",
    "math.SP",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Kernel matrices are of central importance to many applied fields. In this manuscript, we focus on spectral properties of kernel matrices in the so-called \"flat limit\", which occurs when points are close together relative to the scale of the kernel. We establish asymptotic expressions for the determinants of the kernel matrices, which we then leverage to obtain asymptotic expressions for the main terms of the eigenvalues. A separate analysis using Kato's perturbation theory yields expressions for limiting eigenvectors, which are strongly tied to discrete orthogonal polynomials. Both smooth and finitely smooth kernels are covered, with stronger results available in the univariate case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-30T18:09:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A18",
      "47A55",
      "47A75",
      "47B34",
      "60G15",
      "65D05"
    ],
    "keywords": [
      "kernel matrices",
      "flat limit",
      "spectral properties",
      "katos perturbation theory yields expressions",
      "asymptotic expressions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}