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A classical problem that arises in numerous signal processing applicationsasks for the reconstruction of an unknown, $k$-sparse signal $x_0\in R^n$ fromunderdetermined, noisy, linear measurements $y=Ax_0+z\in R^m$. One standardapproach 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 theentries of the sensing matrix $A$ and of the noise vector $z$ are i.i.dGaussian with variances $1/m$ and $\sigma^2$. In the large system limit whenthe problem dimensions grow to infinity, but in constant rates, we\emph{precisely} characterize the limiting behavior of the normalizedsquared-error $\|\hat x-x_0\|^2_2/\sigma^2$. Our numerical illustrationsvalidate our theoretical predictions.

Precise Error Analysis of the $\ell_2$-LASSO