DC algorithm for estimation of sparse Gaussian graphical models.

Sparse estimation of a Gaussian graphical model (GGM) is an important technique for making relationships between observed variables more interpretable. Various methods have been proposed for sparse GGM estimation, including the graphical lasso that uses the ℓ1 norm regularization term, and other methods that use nonconvex regularization terms. Most of these methods approximate the ℓ0 (pseudo) norm by more tractable functions; however, to estimate more accurate solutions, it is preferable to directly use the ℓ0 norm for counting the number of nonzero elements. To this end, we focus on sparse estimation of GGM with the cardinality constraint based on the ℓ0 norm. Specifically, we convert the cardinality constraint into an equivalent constraint based on the largest-K norm, and reformulate the resultant constrained optimization problem into an unconstrained penalty form with a DC (difference of convex functions) representation. To solve this problem efficiently, we design a DC algorithm in which the graphical lasso algorithm is repeatedly executed to solve convex optimization subproblems. Experimental results using two synthetic datasets show that our method achieves results that are comparable to or better than conventional methods for sparse GGM estimation. Our method is particularly advantageous for selecting true edges when cross-validation is used to determine the number of edges. Moreover, our DC algorithm converges within a practical time frame compared to the graphical lasso.