A new approach to Cholesky-based covariance regularization in high dimensions

In this paper we propose a new regression interpretation of the Cholesky factor of the covariance matrix, as opposed to the well-known regression interpretation of the Cholesky factor of the inverse covariance, which leads to a new class of regularized covariance estimators suitable for high-dimensi...

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Bibliographic Details
Published in:Biometrika 2010-09, Vol.97 (3), p.539-550
Main Authors: Rothman, Adam J., Levina, Elizaveta, Zhu, Ji
Format: Article
Language:English
Subjects:
Banding
Cholesky decomposition
Computer applications
Cost estimates
Covariance
Covariance matrices
Data processing
Eigenvalues
Estimating techniques
Estimation methods
Estimators
High dimensional spaces
High-dimensional data
Large p small n
Lasso
Matrix
Maximum likelihood estimation
Maximum likelihood estimators
Maximum likelihood method
Regression analysis
Simulation
Sparsity
Spectral bands
Studies
Variance analysis
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Summary:In this paper we propose a new regression interpretation of the Cholesky factor of the covariance matrix, as opposed to the well-known regression interpretation of the Cholesky factor of the inverse covariance, which leads to a new class of regularized covariance estimators suitable for high-dimensional problems. Regularizing the Cholesky factor of the covariance via this regression interpretation always results in a positive definite estimator. In particular, one can obtain a positive definite banded estimator of the covariance matrix at the same computational cost as the popular banded estimator of Bickel & Levina (2008b), which is not guaranteed to be positive definite. We also establish theoretical connections between banding Cholesky factors of the covariance matrix and its inverse and constrained maximum likelihood estimation under the banding constraint, and compare the numerical performance of several methods in simulations and on a sonar data example.
ISSN:0006-3444
1464-3510
1464-3510
0006-3444
DOI:10.1093/biomet/asq022