The squared-error of generalized LASSO: A precise analysis

We consider the problem of estimating an unknown but structured signal x₀ from its noisy linear observations y = Ax₀ + z ∈ ℝ^m. To the structure of x₀ is associated a structure inducing convex function f(·). We assume that the entries of A are i.i.d. standard normal N(0, 1) and z ~ N(0, σ²I_m). As a measure of performance of an estimate x* of x₀ we consider the “Normalized Square Error” (NSE) ∥x* - x₀∥₂²/σ². For sufficiently small σ, we characterize the exact performance of two different versions of the well known LASSO algorithm. The first estimator is obtained by solving the problem argmin_x ∥y - Ax∥₂ + λf(x). As a function of λ, we identify three distinct regions of operation. Out of them, we argue that “R_ON” is the most interesting one. When λ ∈ R_ON, we show that the NSE is D_f(x₀, λ)/m-D_f(x₀, λ) for small σ, where D_f(x₀, λ) is the expected squared-distance of an i.i.d. standard normal vector to the dilated subdifferential λ · ∂f(x₀). Secondly, we consider the more popular estimator argmin_x 1/2∥y - Ax∥₂². + στ f(x). We propose a formula for the NSE of this estimator by establishing a suitable mapping between this and the previous estimator over the region R_ON. As a useful side result, we find explicit formulae for the optimal estimation performance and the optimal penalty parameters λ* and τ*.