Sparse estimation of large covariance matrices via a nested Lasso penalty

  • Levina E
  • Rothman A
  • Zhu J
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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.

Author-supplied keywords

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

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Authors

  • Elizaveta Levina

  • Adam Rothman

  • Ji Zhu

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