above procedure partially optimizes the bi-convex objective function (\[eqn:obj\]) in two stages: it sets $c_k=1 \;\forall k$, optimizes over $\beta$ and then optimizes over the $c_k$ with $\hat\beta$ fixed. Of course one could iterate these steps in the hopes of obtaining at least a local optimum of the objective function. But we have found that the simple two-step approach works well in practice and is more efficient computationally.

[**Remark B**]{}. The bias induced by setting blocks of the covariance matrix to zero can be seen in a simple example. Let $A$ be a block diagonal matrix with blocks $A_1, A_2$ and let the covariance of the features be $S=A+ \rho ee^T$ where $e$ is a $p$-vector of ones. Assume that $A_1, A_2$ are positive definite. Then by the Sherman-Morrison-Woodbury formula $$\begin{aligned}
S^{-1}=A^{-1} -\frac{\rho A^{-1}ee^TA^{-1}}{1+\rho e^TA^{-1}e}
\label{eqn:smw}\end{aligned}$$ The coefficients for the full least squares fit are $S^{-1}X^Ty$; if instead we set to zero the covariance elements outside of the blocks $A_1, A_2$, the estimates become $A_j^{-1} x_j y$ for $j=1,2$. The second term in (\[eqn:smw\]) represents the bias in using $A^{-1}$ in place of $S^{-1}$, and is generally larger as $\rho$ increases.

Examples {#sec:ex}
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Simulated examples
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In this section, 