Graphical lasso

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In statistics, the graphical lasso^[1] is a penalized likelihood estimator for the precision matrix (also called the concentration matrix or inverse covariance matrix) of a multivariate elliptical distribution. Through the use of an ${\displaystyle L_{1}}$ penalty, it performs regularization to give a sparse estimate for the precision matrix. In the case of multivariate Gaussian distributions, sparsity in the precision matrix corresponds to conditional independence between the variables therefore implying a Gaussian graphical model.

The graphical lasso was originally 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.

Let ${\displaystyle S}$ be the sample covariance matrix of an independent identically distributed sample from a multivariate Gaussian distribution ${\displaystyle X\sim N(\mu ,\Sigma )}$. We are interested in estimating the precision matrix ${\displaystyle \Sigma ^{-1}=\Theta =(\Theta _{ij})}$.

The graphical lasso estimator ${\displaystyle {\hat {\Theta }}}$ is the maximiser of the ${\displaystyle L_{1}}$ penalised log-likelihood:

${\displaystyle {\hat {\Theta }}=\operatorname {argmax} _{\Theta \succ 0}\left(\log \det(\Theta )-\operatorname {tr} (S\Theta )-\lambda \sum _{i,j}|\Theta _{ij}|\right)}$

where ${\displaystyle \lambda }$ is a penalty parameter,^[4] ${\displaystyle \operatorname {tr} }$ is the trace function and ${\displaystyle \Theta \succ 0}$ refers to the set of positive definite matrices.

A popular alternative form of the graphical lasso removes the penalty on the diagonal, only penalising the off-diagonal entries:^[6]

${\displaystyle {\hat {\Theta }}=\operatorname {argmax} _{\Theta \succ 0}\left(\log \det(\Theta )-\operatorname {tr} (S\Theta )-\lambda \sum _{i\neq j}|\Theta _{ij}|\right)}$

Because the graphical lasso estimate is not invariant to scalar multiplication of the variables,^[7] it is important to normalize the data before applying the graphical lasso.

To obtain the estimator in programs, users could use the R package glasso,^[8] GraphicalLasso() class in the scikit-learn Python library,^[9] or the skggm Python package^[10] (similar to scikit-learn).

• Graphical model
• Lasso (statistics)