Sparse Permutation Invariant Covariance Estimation

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. The estimator is required to be positive definite, but we avoid having to use semi-definite programming by re-parameterizing the objective function in terms of the Cholesky factor of the concentration matrix, and derive an iterative optimization algorithm which reduces to solving a linear system at each iteration. Unlike other covariance estimation methods based on the Cholesky factor, our estimator is invariant to variable permutations. The method is compared to other estimators on simulated data and on a real data example of tumor tissue classification using gene expression data.