PRECISE ERROR ANALYSIS OF THE LASSO

A classical problem that arises in numerous signal processing applications asks for the reconstruction of an unknown, k-sparse signal x(0) is an element of R-n from underdetermined, noisy, linear measurements y = Ax(0) + z is an element of R-m. One standard approach is to solve the following convex program (x) over cap = arg min(x) parallel to y - Ax parallel to(2) + lambda parallel to x parallel to(1), which is known as the l(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 precisely characterize the limiting behavior of the normalized squared error parallel to(x) over cap - x(0)parallel to(2)(2)/sigma(2). Our numerical illustrations validate our theoretical predictions.