  We consider a linear regression $y=X\beta+u$ where
$X\in\mathbb{\mathbb{{R}}}^{n\times p}$, $p\gg n,$ and $\beta$ is $s$-sparse.
Motivated by examples in financial and economic data, we consider the situation
where $X$ has highly correlated and clustered columns. To perform sparse
recovery in this setting, we introduce the \emph{clustering removal algorithm}
(CRA), that seeks to decrease the correlation in $X$ by removing the cluster
structure without changing the parameter vector $\beta$. We show that as long
as certain assumptions hold about $X$, the decorrelated matrix will satisfy the
restricted isometry property (RIP) with high probability. We also provide
examples of the empirical performance of CRA and compare it with other sparse
recovery techniques.
