  We consider the problem of recovering a $k$-sparse signal
${\mbox{$\beta$}}_0\in\mathbb{R}^p$ from noisy observations $\bf y={\bf
X}\mbox{$\beta$}_0+{\bf w}\in\mathbb{R}^n$. One of the most popular approaches
is the $l_1$-regularized least squares, also known as LASSO. We analyze the
mean square error of LASSO in the case of random designs in which each row of
${\bf X}$ is drawn from distribution $N(0,{\mbox{$\Sigma$}})$ with general
${\mbox{$\Sigma$}}$. We first derive the asymptotic risk of LASSO in the limit
of $n,p\rightarrow\infty$ with $n/p\rightarrow\delta$. We then examine
conditions on $n$, $p$, and $k$ for LASSO to exactly reconstruct
${\mbox{$\beta$}}_0$ in the noiseless case ${\bf w}=0$. A phase boundary
$\delta_c=\delta(\epsilon)$ is precisely established in the phase space defined
by $0\le\delta,\epsilon\le 1$, where $\epsilon=k/p$. Above this boundary, LASSO
perfectly recovers ${\mbox{$\beta$}}_0$ with high probability. Below this
boundary, LASSO fails to recover $\mbox{$\beta$}_0$ with high probability.
While the values of the non-zero elements of ${\mbox{$\beta$}}_0$ do not have
any effect on the phase transition curve, our analysis shows that $\delta_c$
does depend on the signed pattern of the nonzero values of $\mbox{$\beta$}_0$
for general ${\mbox{$\Sigma$}}\ne{\bf I}_p$. This is in sharp contrast to the
previous phase transition results derived in i.i.d. case with
$\mbox{$\Sigma$}={\bf I}_p$ where $\delta_c$ is completely determined by
$\epsilon$ regardless of the distribution of $\mbox{$\beta$}_0$. Underlying our
formalism is a recently developed efficient algorithm called approximate
message passing (AMP) algorithm. We generalize the state evolution of AMP from
i.i.d. case to general case with ${\mbox{$\Sigma$}}\ne{\bf I}_p$. Extensive
computational experiments confirm that our theoretical predictions are
consistent with simulation results on moderate size system.
