L0 Sparse Inverse Covariance Estimation
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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 l_1 norm. However, the best estimator performance is not always achieved with this penalty. The most natural sparsity promoting "norm" is the non-convex l_0 penalty but its lack of convexity has deterred its use in sparse maximum likelihood estimation. In this paper we consider non-convex l_0 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 l_0 penalty as compared to the l_1 penalty.
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