correlated with each other but not with predictors in other groups.]{} Here, we introduce a method adapted to situations where the sample covariance matrix is approximately block diagonal. Our proposed method, [*the component lasso*]{}, applies a more severe form of decorrelation than the elastic net to exploit this structure.

Consider the inverse of the covariance matrix of the predictors. Zeros in this matrix correspond to conditionally independent variables. Recent work has focused on estimating a sparse version of the inverse covariance by optimizing the $\ell_1$ penalized log-likelihood. The so-called “graphical lasso” algorithm solves the problem by cycling through the variables and fitting a modified lasso regression to each one. In their “scout” procedure, used the graphical lasso in a penalized regression framework to estimate the inverse covariance of $X$. Then they applied a modified form of the lasso to estimate the regression parameters.

More recently, a connection between the graphical lasso and connected components has been established by and . Specifically, the connected components in the estimated inverse covariance matrix correspond exactly to those obtained from single-linkage clustering of the correlation matrix. Clustering the correlated variables before estimating the parameters has been suggested by and .

In this paper, we 