Universality of the elastic net error

We consider the problem of reconstructing a vector x
₀
ϵ ℝ
ⁿ
from noisy linear observations y = Ax
₀
+ w, where A ϵ ℝ
^m×n
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→∞ and m/n → δ > 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.