Precise error analysis of the LASSO

A classical problem that arises in numerous signal processing applications asks for the reconstruction of an unknown, k-sparse signal x₀ ∈ ℝⁿ from underdetermined, noisy, linear measurements y = Ax₀ + z ∈ ℝ^m. One standard approach is to solve the following convex program x̂ = arg min_x ∥y - Ax∥₂+λ∥x∥₁, which is known as the ℓ₂-LASSO. We assume that the entries of the sensing matrix A and of the noise vector z are i.i.d Gaussian with variances 1/m and σ². In the large system limit when the problem dimensions grow to infinity, but in constant rates, we precisely characterize the limiting behavior of the normalized squared error ∥x̂ - x₀∥₂²/σ². Our numerical illustrations validate our theoretical predictions.