Precise Error Analysis of the $\ell_2$-LASSO

A classical problem that arises in numerous signal processing applications asks for the reconstruction of an unknown, $k$-sparse signal $x_0\in R^n$ from underdetermined, noisy, linear measurements $y=Ax_0+z\in R^m$. One standard approach is to solve the following convex program $\hat x=\arg\min_x \|y-Ax\|_2 + \lambda \|x\|_1$, which is known as the $\ell_2$-LASSO. We assume that the entries of the sensing matrix $A$ and of the noise vector $z$ are i.i.d Gaussian with variances $1/m$ and $\sigma^2$. In the large system limit when the problem dimensions grow to infinity, but in constant rates, we \emph{precisely} characterize the limiting behavior of the normalized squared-error $\|\hat x-x_0\|^2_2/\sigma^2$. Our numerical illustrations validate our theoretical predictions.