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 l1β
norm. However, the best estimator performance is not always achieved with this
penalty. The most natural sparsity promoting "norm" is the non-convex l0β
penalty but its lack of convexity has deterred its use in sparse maximum
likelihood estimation. In this paper we consider non-convex l0β 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 non-trivial, and we demonstrate via simulations the reduced
bias and superior quality of the l0β penalty as compared to the l1β
penalty