Sparse estimation of large covariance matrices via a hierarchical Lasso penalty

Ji Zhu
University of Michigan

The paper proposes a new covariance estimator for large covariance matrices. Using the Cholesky decomposition of the inverse, we impose a banded structure on the Cholesky factor, and select the band width adaptively for each row of the Cholesky factor, using a novel penalty we call hierarchical 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. A maximum likelihood a nd a least squares variants are developed, and an iterative algorithm for optimization is developed. The estimator is compared to a number of other covariance estimators and is shown to do best, both in simulations and on two real data examples. Simulations show that the margin by which the estimator outperforms its competitors tends to increase with dimension.

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