## Sparse permutation invariant covariance estimation.(English)Zbl 1320.62135

Summary: 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.

### MSC:

 62H20 Measures of association (correlation, canonical correlation, etc.) 62H12 Estimation in multivariate analysis

glasso; pcalg
Full Text:

### References:

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