Universality Of The Elastic Net Error

We consider the problem of reconstructing a vector x(0) is an element of R-n from noisy linear observations y = Ax(0) + w, where A is an element of R-mxn is a known operator and w is a noise vector, using the elastic net method. Assuming that A is random with independent and identically distributed entries, and under suitable moment conditions, we prove the following universality result. In the high-dimensional asymptotics n -> infinity and m/n -> delta > 0, the normalized error of the elastic net minimizer converges in probability to a limit, that does not depend on the exact distribution that the entries are drawn from. We also provide an explicit formula for the limit.