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The graphical lasso (glasso) is a widely-used fast algorithm for estimatingsparse inverse covariance matrices. The glasso solves an L1 penalized maximumlikelihood problem and is available as an R library on CRAN. The output fromthe glasso, a regularized covariance matrix estimate a sparse inversecovariance matrix estimate, not only identify a graphical model but can alsoserve as intermediate inputs into multivariate procedures such as PCA, LDA,MANOVA, and others. The glasso indeed produces a covariance matrix estimatewhich solves the L1 penalized optimization problem in a dual sense; however,the method for producing the inverse covariance matrix estimator after thisoptimization is inexact and may produce asymmetric estimates. This problem isexacerbated when the amount of L1 regularization that is applied is small,which in turn is more likely to occur if the true underlying inverse covariancematrix is not sparse. The lack of symmetry can potentially have consequences.First, it implies that the covariance and inverse covariance estimates are notnumerical inverses of one another, and second, asymmetry can possibly lead tonegative or complex eigenvalues,rendering many multivariate procedures whichmay depend on the inverse covariance estimator unusable. We demonstrate thisproblem, explain its causes, and propose possible remedies.

A note on the lack of symmetry in the graphical lasso