Electronic Journal of Statistics

Sparse permutation invariant covariance estimation

Adam J. Rothman, Peter J. Bickel, Elizaveta Levina, and Ji Zhu

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The paper proposes a method for constructing a sparse estimator for the inverse covariance (concentration) matrix in high-dimensional settings. The estimator uses a penalized normal likelihood approach and forces sparsity by using a lasso-type penalty. We establish a rate of convergence in the Frobenius norm as both data dimension p and sample size n are allowed to grow, and show that the rate depends explicitly on how sparse the true concentration matrix is. We also show that a correlation-based version of the method exhibits better rates in the operator norm. We also derive a fast iterative algorithm for computing the estimator, which relies on the popular Cholesky decomposition of the inverse but produces a permutation-invariant estimator. The method is compared to other estimators on simulated data and on a real data example of tumor tissue classification using gene expression data.

Article information

Electron. J. Statist., Volume 2 (2008), 494-515.

First available in Project Euclid: 26 June 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H20: Measures of association (correlation, canonical correlation, etc.)
Secondary: 62H12: Estimation

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


Rothman, Adam J.; Bickel, Peter J.; Levina, Elizaveta; Zhu, Ji. Sparse permutation invariant covariance estimation. Electron. J. Statist. 2 (2008), 494--515. doi:10.1214/08-EJS176. https://projecteuclid.org/euclid.ejs/1214491853

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