The Noise-Sensitivity Phase Transition in Compressed Sensing

Consider the noisy underdetermined system of linear equations: y = Ax
₀
+ z, with A an n × N measurement matrix, n <; N, and z ~ N(0, σ
²
I) a Gaussian white noise. Both y and A are known, both x
₀
and z are unknown, and we seek an approximation to x
₀
. When x
₀
has few nonzeros, useful approximations are often obtained by ℓ
₁
-penalized ℓ
₂
minimization, in which the reconstruction x̂
^1,λ
solves min{||y - Ax||
₂
²
/2 + λ||x||
₁
}. Consider the reconstruction mean-squared error MSE = E|| x̂
^1,λ
- x
₀
||
₂
²
/N, and define the ratio MSE/σ
²
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 ℓ
₁
- ℓ
₀
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 ℓ
₁
-penalized reconstruction. The focus of the present paper is on computing the minimax formal MSE within the class of sparse signals x
⁰
.