The squared-error of generalized LASSO: A precise analysis

Abstract

We consider the problem of estimating an unknown but structured signal x0 from its noisy linear observations y = Ax_0 + 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, σ^2I_m). As a measure of performance of an estimate x^* of x_0 we consider the "Normalized Square Error" (NSE) ∥x* - x0∥^2_2/σ^2. 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 arg min_x ∥y - Ax∥_2 + λf(x). As a function of λ, we identify three distinct regions of operation. Out of them, we argue that "RON" is the most interesting one. When λ ∈ R_(ON), we show that the NSE is (D_f(x_0, λ))/(m-D_f(x_0, λ)) for small σ, where D_f(x_0, λ) is the expected squared-distance of an i.i.d. standard normal vector to the dilated subdifferential λ · ∂f(x_0). Secondly, we consider the more popular estimator arg min_x 1/2∥y - Ax∥^2_2 + 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 RON. As a useful side result, we find explicit formulae for the optimal estimation performance and the optimal penalty parameters λ* and τ*.

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