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 ℓ1 penalized maximum likelihood problem and is available as an R library on CRAN. The output from the glasso, a regularized covariance matrix estimate Σˆglasso and a sparse inverse covariance matrix estimate Ωˆglasso, not only identify a graphical model but can also serve as intermediate inputs into multivariate procedures such as PCA, LDA, MANOVA, and others. The glasso indeed produces a covariance matrix estimate Σˆglasso which solves the ℓ1 penalized optimization problem in a dual sense; however, the method for producing Ωˆglasso after this optimization is inexact and may produce asymmetric estimates. This problem is exacerbated when the amount of ℓ1 regularization that is applied is small, which in turn is more likely to occur if the true underlying inverse covariance matrix is not sparse. The lack of symmetry can potentially have consequences. First, it implies that Σˆglasso−1≠Ωˆglasso and, second, asymmetry can possibly lead to negative or complex eigenvalues, rendering many multivariate procedures which may depend on Ωˆglasso unusable. We demonstrate this problem, explain its causes, and propose possible remedies.
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