Sparse estimation of large covariance matrices via a nested Lasso penalty

Elizaveta Levina, Adam Rothman, Ji Zhu

Research output: Contribution to journalArticlepeer-review

132 Scopus citations

Abstract

The paper proposes a new covariance estimator for large covariance matrices when the variables have a natural ordering. Using the Cholesky decomposition of the inverse, we impose a banded structure on the Cholesky factor, and select the bandwidth adaptively for each row of the Cholesky factor, using a novel penalty we call nested Lasso. This structure has more flexibility than regular banding, but, unlike regular Lasso applied to the entries of the Cholesky factor, results in a sparse estimator for the inverse of the covariance matrix. An iterative algorithm for solving the optimization problem is developed. The estimator is compared to a number of other covariance estimators and is shown to do best, both in simulations and on a real data example. Simulations show that the margin by which the estimator outperforms its competitors tends to increase with dimension.

Original languageEnglish (US)
Pages (from-to)245-263
Number of pages19
JournalAnnals of Applied Statistics
Volume2
Issue number1
DOIs
StatePublished - Mar 2008

Keywords

  • Cholesky decomposition
  • Covariance matrix
  • High dimension low sample size
  • Large p small n
  • Lasso
  • Sparsity

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