High-dimensional covariance estimation based on Gaussian graphical models

Shuheng Zhou, Philipp Rütimami, Min Xu, Peter Bühlmann

Research output: Contribution to journalArticlepeer-review

48 Scopus citations


Undirected graphs are often used to describe high dimensional distributions. Under sparsity conditions, the graph can be estimated using l1-penalization methods. We propose and study the following method. We combine a multiple regression approach with ideas of thresholding and refitting: first we infer a sparse undirected graphical model structure via thresholding of each among many l1-norm penalized regression functions; we then estimate the covariance matrix and its inverse using the maximum likelihood estimator. We show that under suitable conditions, this approach yields consistent estimation in terms of graphical structure and fast convergence rates with respect to the operator and Frobenius norm for the covariance matrix and its inverse. We also derive an explicit bound for the Kullback Leibler divergence.

Original languageEnglish (US)
Pages (from-to)2975-3026
Number of pages52
JournalJournal of Machine Learning Research
StatePublished - Oct 2011
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Software
  • Control and Systems Engineering
  • Statistics and Probability
  • Artificial Intelligence


  • Covariance estimation
  • Graphical model selection
  • Lasso
  • Nodewise regression
  • Thresholding


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