Graphical lasso

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In statistics, the graphical lasso^[1] is a sparse penalized maximum likelihood estimator for the concentration or precision matrix (inverse of covariance matrix) of a multivariate elliptical distribution. The original variant was formulated to solve Dempster's covariance selection problem^[2][3] for the multivariate Gaussian distribution when observations were limited. Subsequently, the optimization algorithms to solve this problem were improved^[4] and extended^[5] to other types of estimators and distributions.

Consider observations ${\displaystyle X_{1},X_{2},\ldots ,X_{n}}$ from multivariate Gaussian distribution ${\displaystyle X\sim N(0,\Sigma )}$. We are interested in estimating the precision matrix ${\displaystyle \Theta =\Sigma ^{-1}}$.

The graphical lasso estimator is the ${\displaystyle {\hat {\Theta }}}$ such that:

${\displaystyle {\hat {\Theta }}=\operatorname {argmin} _{\Theta \geq 0}\left(\operatorname {tr} (S\Theta )-\log \det(\Theta )+\lambda \sum _{j\neq k}|\Theta _{jk}|\right)}$

where ${\displaystyle S}$ is the sample covariance, and ${\displaystyle \lambda }$ is the penalizing parameter.^[4]

To obtain the estimator in programs, users could use the R package glasso,^[6] GraphicalLasso() class in the scikit-learn Python library,^[7] or the skggm Python package^[8] (similar to scikit-learn).

• Graphical model
• Lasso (statistics)