A note on the lack of symmetry in the graphical lasso

The graphical lasso (glasso) is a widely-used fast algorithm for estimating sparse inverse covariance matrices. The glasso solves an L_1 penalized maximum likelihood problem and is implemented on CRAN. The output from the glasso, a regularized covariance matrix estimate Sigma_glasso and a sparse inverse covariance matrix estimate Omega_glasso, not only identify a graphical model but can also serve as intermediate inputs into multivariate procedures such as PCA, LDA, MANOVA, and others. Despite its strengths, the glasso may produce asymmetric estimates Omega_glasso, a problem which is exacerbated when the L_1 regularization applied is small. This is more likely to occur if the true underlying inverse covariance matrix is not so sparse. The lack of symmetry can potentially have consequences. First, it implies that (Sigma_glasso)^(-1) is not equal to Omega_glasso and second, asymmetry can possibly lead to negative or complex eigenvalues, rendering many multivariate procedures which may depend on Omega_glasso unusable. We demonstrate this problem, explain its causes, and propose possible remedies.