Asymptotically exact error analysis for the generalized equation-LASSO

Given an unknown signal x
₀
∈ R
ⁿ
and linear noisy measurements y = Ax
₀
+ σv ∈ R
^m
, the generalized ℓ
₂
²
-LASSO solves x̂ := arg min
^x
1/2∥y-Ax ∥
₂
²
+ σλf(x). Here, f is a convex regularization function (e.g. ℓ
₁
-norm, nuclearnorm) aiming to promote the structure of x
₀
(e.g. sparse, lowrank), and, λ ≥ 0 is the regularizer parameter. A related optimization problem, though not as popular or well-known, is often referred to as the generalized ℓ
₂
-LASSO and takes the form x̂̂̂ := arg min
_x
∥y-Ax∥
₂
+λf(x), and has been analyzed by Oymak, Thrampoulidis and Hassibi. Oymak et al. further made conjectures about the performance of the generalized ℓ
₂
²
-LASSO. This paper establishes these conjectures rigorously. We measure performance with the normalized squared error NSE(σ) := ∥x-x
₀
∥
₂
²
/(mσ
²
). Assuming the entries of A are i.i.d. Gaussian N (0,1/m) and those of v are i.i.d. N(0,1), we precisely characterize the “asymptotic NSE” aNSE := lim
_σ→0
NSE(σ) when the problem dimensions tend to infinity in a proportional manner. The role of λ, f and x
₀
is explicitly captured in the derived expression via means of a single geometric quantity, the Gaussian distance to the subdifferential. We conjecture that aNSE = sup
_σ>0
NSE(σ). We include detailed discussions on the interpretation of our result, make connections to relevant literature and perform computational experiments that validate our theoretical findings.