The Noise-Sensitivity Phase Transition in Compressed Sensing

Consider the noisy underdetermined system of linear equations: y = Ax₀ + z, with A an n × N measurement matrix, n <; N, and z ~ N(0, σ²I) a Gaussian white noise. Both y and A are known, both x₀ and z are unknown, and we seek an approximation to x₀. When x₀ has few nonzeros, useful approximations are often obtained by ℓ₁-penalized ℓ₂ minimization, in which the reconstruction x̂^1,λ solves min{||y - Ax||₂²/2 + λ||x||₁}. Consider the reconstruction mean-squared error MSE = E|| x̂^1,λ - x₀||₂²/N, and define the ratio MSE/σ² as the noise sensitivity. Consider matrices A with i.i.d. Gaussian entries and a large-system limit in which n, N → ∞ with n/N → δ and k/n → ρ. We develop exact expressions for the asymptotic MSE of x̂^1,λ , and evaluate its worst-case noise sensitivity over all types of k-sparse signals. The phase space 0 ≤ 8, ρ ≤ 1 is partitioned by the curve ρ = ρ_MSE(δ) into two regions. Formal noise sensitivity is bounded throughout the region ρ = ρ_MSE(δ) and is unbounded throughout the region ρ = ρ_MSE(δ). The phase boundary ρ = ρ_MSE(δ) is identical to the previously known phase transition curve for equivalence of ℓ₁ - ℓ₀ minimization in the k-sparse noiseless case. Hence, a single phase boundary describes the fundamental phase transitions both for the noise less and noisy cases. Extensive computational experiments validate these predictions, including the existence of game-theoretical structures underlying it (saddlepoints in the payoff, - - least-favorable signals and maximin penalization). Underlying our formalism is an approximate message passing soft thresholding algorithm (AMP) introduced earlier by the authors. Other papers by the authors detail expressions for the formal MSE of AMP and its close connection to ℓ₁-penalized reconstruction. The focus of the present paper is on computing the minimax formal MSE within the class of sparse signals x⁰.