{
  "id": "2404.12589",
  "version": "v1",
  "published": "2024-04-19T02:35:03.000Z",
  "updated": "2024-04-19T02:35:03.000Z",
  "title": "A rate-distortion framework for MCMC algorithms: geometry and factorization of multivariate Markov chains",
  "authors": [
    "Michael C. H. Choi",
    "Youjia Wang",
    "Geoffrey Wolfer"
  ],
  "comment": "63 pages, 6 figures",
  "categories": [
    "math.PR",
    "cs.IT",
    "math.IT",
    "math.OC",
    "stat.CO"
  ],
  "abstract": "We introduce a framework rooted in a rate distortion problem for Markov chains, and show how a suite of commonly used Markov Chain Monte Carlo (MCMC) algorithms are specific instances within it, where the target stationary distribution is controlled by the distortion function. Our approach offers a unified variational view on the optimality of algorithms such as Metropolis-Hastings, Glauber dynamics, the swapping algorithm and Feynman-Kac path models. Along the way, we analyze factorizability and geometry of multivariate Markov chains. Specifically, we demonstrate that induced chains on factors of a product space can be regarded as information projections with respect to a particular divergence. This perspective yields Han--Shearer type inequalities for Markov chains as well as applications in the context of large deviations and mixing time comparison.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-19T02:35:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F10",
      "60J10",
      "60J22",
      "94A15",
      "94A17"
    ],
    "keywords": [
      "multivariate markov chains",
      "rate-distortion framework",
      "mcmc algorithms",
      "perspective yields han-shearer type inequalities",
      "markov chain monte carlo"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 63,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}