Precise Error Analysis of the $\ell_2$-LASSO

TL;DR

This paper provides a precise theoretical analysis of the error behavior of the $ ext{l}_2$-LASSO method for reconstructing sparse signals from noisy measurements, validated by numerical experiments.

Contribution

It offers an exact characterization of the normalized squared-error of $ ext{l}_2$-LASSO in the large system limit with Gaussian sensing matrices and noise.

Findings

The normalized squared-error converges to a deterministic limit.

Theoretical predictions match numerical simulations.

Provides insights into the error behavior in high-dimensional settings.

Abstract

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.