Precise Error Analysis of the `2-lasso

A classical problem that arises in numerous signal processing applications asks for the reconstruction of an unknown, ksparse signal x0 ∈ R from underdetermined, noisy, linear measurements y = Ax0 + z ∈ R. One standard approach is to solve the following convex program x̂ = arg minx ‖y− Ax‖2+λ‖x‖1, which is known as the `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 σ. In the large system limit when the problem dimensions grow to infinity, but in constant rates, we precisely characterize the limiting behavior of the normalized squared error ‖x̂− x0‖2/σ. Our numerical illustrations validate our theoretical predictions.