right). The signal variables are shown in blue, while the non-signal variables are in red.*[]{data-label="fig:paths"}](cl_path.png){width="\linewidth"}

The example shows the role that NNLS plays in selecting the relevant component which contains the signal variables (in blue) and reducing the coefficients of the noise variables (in red) in the second component to zero. This illustrates the possible improvements that can be achieved by finding the block-diagonal structure of the sample covariance matrix, as compared to standard methods.

The remainder of the paper is organized as follows. We explain our algorithm in Section \[sec:comp\_lass\]. Section \[sec:ex\] includes simulated and real data results. Section \[sec:comp\] focuses on the computational complexity of the component lasso, and presents ideas for making it more efficient. We conclude the paper with a short discussion in Section \[sec:discussion\], including possible extensions to generalized linear models.

The Component Lasso {#sec:comp_lass}
===================

The main idea
-------------

The lasso minimizes the $\ell_1$ penalized criterion (\[eqn:lasso\]) whose corresponding subgradient equation is $$X^TX\beta - X^T y + \lambda\cdot \text{sign}(\beta) = 0,
\label{eqn:subg}$$ where $\text{sign}(\beta)$ is a vector with components $s_j=\text{sign} (\beta_j)$ if $\beta_j\neq 0$ and $s_j \in [-1,1]$ if $\beta_j= 0$.

The solution to the lasso can be written as $$\hat{\beta} = (X^TX)^{-}(X^Ty-\lambda\cdot  \text{sign}(\hat{\beta}))
