\label{eqn:lassosol}$$ where $(X^TX)^{-}$ represents a generalized inverse of $X^TX$.

Let $\Sigma=\text{cov}(X)$. We propose replacing $(X^TX)^{-}$ by a block diagonal estimate $n^{-1}\hat\Theta \approx n^{-1}\Sigma^{-1} $, the blocks of $\hat\Theta$ being the (estimated) connected components. Finding $K$ connected components splits the subgradient equation into $K$ separate equations: $$\label{eqn:subg2}
    X^T_k X_k \beta_k - X_k^T y + \lambda\cdot \text{sign}(\beta_k ) = 0$$ for $k = 1, 2,\dots K$, where $X_k$ is a subset of $X$ containing the observations of the predictors in the $k$th component, and $\beta_k$ contains the corresponding coefficients.

Each subproblem can be solved individually using a standard lasso or elastic net algorithm. The resultant coefficients $\beta_k$ are then combined into a solution to the original problem. The use of the block-diagonal covariance matrix creates a substantial bias in the coefficient estimates, so the combination step is quite important. We scale the componentwise solution vectors $\hat\beta_1, \hat\beta_2, \ldots \hat\beta_K$ using a non-negative least squares refitting of $y$ on $\{\hat{y}_k=X_k\hat\beta_k\}, k=1, \dots, K$. The non-negativity constraint seems natural since each componentwise predictor should have positive correlation with the outcome.

The component lasso objective function, corresponding to a block diagonal estimate of the sample covariance with connected components $C_1,\dots,C_K$ is: $$\begin{aligned}
J(\beta, c)= \sum_ 