SCAD-Penalized Complex Gaussian Graphical Model Selection

We consider the problem of estimating the conditional independence graph (CIG) of a sparse, high-dimensional proper complex-valued Gaussian graphical model (CGGM). For CGGMs, the problem reduces to estimation of the inverse covariance matrix with more unknowns than the sample size. We consider a smoothly clipped absolute deviation (SCAD) penalty instead of the $\ell_1$-penalty to regularize the problem, and analyze a SCAD-penalized log-likelihood based objective function to establish consistency and sparsistency of a local estimator of inverse covariance in a neighborhood of the true value. A numerical example is presented to illustrate the advantage of SCAD-penalty over the usual $\ell_1$-penalty