{
  "id": "2307.11632",
  "version": "v1",
  "published": "2023-07-21T14:55:50.000Z",
  "updated": "2023-07-21T14:55:50.000Z",
  "title": "Matrix concentration inequalities with dependent summands and sharp leading-order terms",
  "authors": [
    "Alexander Van Werde",
    "Jaron Sanders"
  ],
  "comment": "69 pages, 4 figures",
  "categories": [
    "math.PR",
    "math.OA"
  ],
  "abstract": "We establish sharp concentration inequalities for sums of dependent random matrices. Our results concern two models. First, a model where summands are generated by a $\\psi$-mixing Markov chain. Second, a model where summands are expressed as deterministic matrices multiplied by scalar random variables. In both models, the leading-order term is provided by free probability theory. This leading-order term is often asymptotically sharp and, in particular, does not suffer from the logarithmic dimensional dependence which is present in previous results such as the matrix Khintchine inequality. A key challenge in the proof is that techniques based on classical cumulants, which can be used in a setting with independent summands, fail to produce efficient estimates in the Markovian model. Our approach is instead based on Boolean cumulants and a change-of-measure argument. We discuss applications concerning community detection in Markov chains, random matrices with heavy-tailed entries, and the analysis of random graphs with dependent edges.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-21T14:55:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "60J05",
      "46L53"
    ],
    "keywords": [
      "matrix concentration inequalities",
      "sharp leading-order terms",
      "dependent summands",
      "markov chain",
      "scalar random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 69,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}