{
  "id": "1404.0788",
  "version": "v3",
  "published": "2014-04-03T07:41:08.000Z",
  "updated": "2015-01-16T17:46:10.000Z",
  "title": "On the principal components of sample covariance matrices",
  "authors": [
    "Alex Bloemendal",
    "Antti Knowles",
    "Horng-Tzer Yau",
    "Jun Yin"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "We introduce a class of $M \\times M$ sample covariance matrices $\\mathcal Q$ which subsumes and generalizes several previous models. The associated population covariance matrix $\\Sigma = \\mathbb E \\cal Q$ is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of $\\Sigma - I_M$ may depend on $M$ in an arbitrary fashion. We investigate the principal components, i.e.\\ the top eigenvalues and eigenvectors, of $\\mathcal Q$. We derive precise large deviation estimates on the generalized components $\\langle \\mathbf w, \\boldsymbol \\xi_i \\rangle$ of the outlier and non-outlier eigenvectors $\\boldsymbol \\xi_i$. Our results also hold near the so-called BBP transition, where outliers are created or annihilated, and for degenerate or near-degenerate outliers. We believe the obtained rates of convergence to be optimal. In addition, we derive the asymptotic distribution of the generalized components of the non-outlier eigenvectors. A novel observation arising from our results is that, unlike the eigenvalues, the eigenvectors of the principal components contain information about the \\emph{subcritical} spikes of $\\Sigma$. The proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix $\\mathcal Q$, satisfying $\\mathbb E \\mathcal Q = I_M$, as input: the isotropic local Marchenko-Pastur law established in [9], level repulsion, and quantum unique ergodicity of the eigenvectors. The latter is a special case of a new universality result for the joint eigenvalue-eigenvector distribution.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-23T16:02:27.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-01-16T17:46:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B52",
      "60B20",
      "82B44"
    ],
    "keywords": [
      "sample covariance matrices",
      "isotropic local marchenko-pastur law",
      "non-outlier eigenvectors",
      "derive precise large deviation estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.0788B"
    }
  }
}