$l_{0}$ Sparse Inverse Covariance Estimation

Recently, there has been focus on penalized log-likelihood covariance estimation for sparse inverse covariance (precision) matrices. The penalty is responsible for inducing sparsity, and a very common choice is the convex l₁ norm. However, the best estimator performance is not always achieved with this penalty. The most natural sparsity promoting “norm” is the nonconvex l₀ penalty but its lack of convexity has deterred its use in sparse maximum likelihood estimation. In this paper, we consider nonconvex l₀ penalized log-likelihood inverse covariance estimation and present a novel cyclic descent algorithm for its optimization. Convergence to a local minimizer is proved, which is highly nontrivial, and we demonstrate via simulations the reduced bias and superior quality of the l₀ penalty as compared to the l₁ penalty.