  This paper concerns the performance of the LASSO (also knows as basis pursuit
denoising) for recovering sparse signals from undersampled, randomized, noisy
measurements. We consider the recovery of the signal $x_o \in \mathbb{R}^N$
from $n$ random and noisy linear observations $y= Ax_o + w$, where $A$ is the
measurement matrix and $w$ is the noise. The LASSO estimate is given by the
solution to the optimization problem $x_o$ with $\hat{x}_{\lambda} = \arg
\min_x \frac{1}{2} \|y-Ax\|_2^2 + \lambda \|x\|_1$. Despite major progress in
the theoretical analysis of the LASSO solution, little is known about its
behavior as a function of the regularization parameter $\lambda$. In this paper
we study two questions in the asymptotic setting (i.e., where $N \rightarrow
\infty$, $n \rightarrow \infty$ while the ratio $n/N$ converges to a fixed
number in $(0,1)$): (i) How does the size of the active set
$\|\hat{x}_\lambda\|_0/N$ behave as a function of $\lambda$, and (ii) How does
the mean square error $\|\hat{x}_{\lambda} - x_o\|_2^2/N$ behave as a function
of $\lambda$? We then employ these results in a new, reliable algorithm for
solving LASSO based on approximate message passing (AMP).
