Graphical lasso

In statistics, the graphical lasso is an algorithm to estimate the precision matrix (inverse of covariance matrix) from the observations from multivariate Gaussian distribution.

Setting

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

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

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

where $S$ is the sample covariance, and $\lambda$ is the penalizing parameter.

Application

To obtain the estimator in programs, users could use the R package glasso,, GraphicalLasso() class in Python Scikit-Learn package, or the skggm Python package (similar to scikit-learn).