propose a new simple idea to make use of the connected components in penalized regression. The [*component lasso*]{} works by (a) finding the connected components of the estimated covariance matrix, (b) solving separate lasso problems for each component, and then (c) combining the componentwise predictions into one final prediction. We show that this approach can improve the accuracy, and interpretability of the lasso and elastic net methods. The method is summarized in Figure \[fig:clgph\].

![*The component lasso steps: The predictors are split according to the estimated connected components of the sample covariance matrix. The lasso is applied to each subset of predictors to separately estimate the coefficients and to predict the response. Finally, the different coefficient vectors are combined using a non-negative least squares fit of $y$ on the $K$ predictions from each component.* []{data-label="fig:clgph"}](compgraph.png)

The following example motivates the remainder of the paper. Consider eight predictors, and let the corresponding covariance matrix be block diagonal with two blocks. Suppose that the predictors corresponding to the first block, or equivalently component, are all signal variables. The second component only contains noise variables. Figure \[fig:paths\] shows the coefficient paths for the naive and non naive elastic net, and the component lasso 