{
  "id": "2306.07255",
  "version": "v1",
  "published": "2023-06-12T17:25:12.000Z",
  "updated": "2023-06-12T17:25:12.000Z",
  "title": "Conditional Matrix Flows for Gaussian Graphical Models",
  "authors": [
    "Marcello Massimo Negri",
    "F. Arend Torres",
    "Volker Roth"
  ],
  "categories": [
    "cs.LG",
    "stat.ML"
  ],
  "abstract": "Studying conditional independence structure among many variables with few observations is a challenging task. Gaussian Graphical Models (GGMs) tackle this problem by encouraging sparsity in the precision matrix through an $l_p$ regularization with $p\\leq1$. However, since the objective is highly non-convex for sub-$l_1$ pseudo-norms, most approaches rely on the $l_1$ norm. In this case frequentist approaches allow to elegantly compute the solution path as a function of the shrinkage parameter $\\lambda$. Instead of optimizing the penalized likelihood, the Bayesian formulation introduces a Laplace prior on the precision matrix. However, posterior inference for different $\\lambda$ values requires repeated runs of expensive Gibbs samplers. We propose a very general framework for variational inference in GGMs that unifies the benefits of frequentist and Bayesian frameworks. Specifically, we propose to approximate the posterior with a matrix-variate Normalizing Flow defined on the space of symmetric positive definite matrices. As a key improvement on previous work, we train a continuum of sparse regression models jointly for all regularization parameters $\\lambda$ and all $l_p$ norms, including non-convex sub-$l_1$ pseudo-norms. This is achieved by conditioning the flow on $p>0$ and on the shrinkage parameter $\\lambda$. We have then access with one model to (i) the evolution of the posterior for any $\\lambda$ and for any $l_p$ (pseudo-) norms, (ii) the marginal log-likelihood for model selection, and (iii) we can recover the frequentist solution paths as the MAP, which is obtained through simulated annealing.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-12T17:25:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gaussian graphical models",
      "conditional matrix flows",
      "shrinkage parameter",
      "precision matrix",
      "frequentist solution paths"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}