  We study the problem of estimating a $k$-sparse signal
${\mbox{$\beta$}}_0\in{\bf R}^p$ from a set of noisy observations ${\bf
y}\in{\bf R}^n$ under the model ${\bf y}={\bf X}{\mbox{$\beta$}}+{\bf w}$,
where ${\bf X}\in{\bf R}^{n\times p}$ is the measurement matrix the row of
which is drawn from distribution $N(0,{\mbox{$\Sigma$}})$. We consider the
class of $L_q$-regularized least squares (LQLS) given by the formulation
$\hat{\mbox{$\beta$}}(\lambda,q)=\text{argmin}_{{\mbox{$\beta$}}\in{\bf
R}^p}\frac{1}{2}\|{\bf y}-{\bf
X}{\mbox{$\beta$}}\|^2_2+\lambda\|{\mbox{$\beta$}}\|_q^q$, where $\|\cdot\|_q$
$(0\le q\le 2)$ denotes the $L_q$-norm. In the setting $p,n,k\rightarrow\infty$
with fixed $k/p=\epsilon$ and $n/p=\delta$, we derive the asymptotic risk of
$\hat{\mbox{$\beta$}}(\lambda,q)$ for arbitrary covariance matrix
${\mbox{$\Sigma$}}$ which generalizes the existing results for standard
Gaussian design, i.e. $X_{ij}\overset{i.i.d}{\sim}N(0,1)$. We perform a
higher-order analysis for LQLS in the small-error regime in which the first
dominant term can be used to determine the phase transition behavior of LQLS.
Our results show that the first dominant term does not depend on the covariance
structure of ${\mbox{$\Sigma$}}$ in the cases $0\le q< 1$ and $1< q\le 2$ which
indicates that the correlations among predictors only affect the phase
transition curve in the case $q=1$ a.k.a. LASSO. To study the influence of the
covariance structure of ${\mbox{$\Sigma$}}$ on the performance of LQLS in the
cases $0\le q< 1$ and $1<q\le 2$, we derive the explicit formulas for the
second dominant term in the expansion of the asymptotic risk in terms of small
error. Extensive computational experiments confirm that our analytical
predictions are consistent with numerical results.
