cut the dendrogram at level $\tau$ [@TWS2013]. Single linkage clustering is sometimes not very attractive in practice, since it can produce long and stringy clusters and hence components of very unequal size. However, these same authors show that under regularity conditions on $S$, application of average or complete linkage agglomerative clustering also consistently estimates the connected components. Hence we are free to use average, single or complete linkage clustering; we use average linkage in the examples of this paper.

Summary of the component lasso algorithm
----------------------------------------

1.  Apply average, single or complete linkage clustering to $S=X^TX/n$ and cut the dendrogram at level $\tau$ to produce components $C_1, C_2, \ldots C_K$.

2.  For each component $k=1,2,\ldots K$ and fixed elastic net parameter $\alpha$, compute a path of elastic net solutions $\hat \beta_{k,\alpha,\tau}(\lambda)$ over a grid of $\lambda$ values. Let $\hat y_{k,\alpha,\tau}(\lambda)$ be the predicted values from the $k$th fit.

3.  Compute the non-negative least squares (NNLS) fit of $y$ on $\{\hat y_{1,\alpha,\tau}(\lambda), \hat y_{2,\alpha,\tau}(\lambda), \ldots \hat y_{K,\alpha,\tau}(\lambda)\}$, yielding weights $\{\hat c_1, \hat c_2, \ldots \hat c_K\}$. Finally, form the overall estimate $\hat\beta_{\alpha,\tau}(\lambda)  
    =\sum_{k=1}^K \hat c_k \hat \beta_{k,\alpha,\tau}(\lambda)$.

4.  Estimate optimal values of $\tau, \alpha$ and $\lambda$ by cross-validation.

[**Remark A**]{}. The 