Phase transition and noise sensitivity of ℓp-minimization for 0 ≤ p ≤ 1

Recovering a sparse vector x
₀
∈ ℝ
^N
from its noisy linear observations, y ∈ ℝ
ⁿ
with y = Ax
₀
+ w, has been the central problem of compressed sensing. One of the classes of recovery algorithms that has attracted attention is the class of ℓ
_p
-regularized least squares (LPLS) that seeks the minimum of 1/2 ∥y - Ax∥
₂
²
+ λ∥x∥
_p
^p
for p ∈ [0, 1]. In this paper we employ the Replica method
¹
from statistical physics to analyze the global minima of LPLS. Our paper reveals several surprising asymptotic properties of LPLS: (i) The phase transition curve of LPLS is the same for every 0 ≤ p <; 1. These phase transition curves are much higher than the phase transition curve for p = 1. (ii) The phase transition curve of LPLS for every value of 0 ≤ p ≤ 1 depends only on the sparsity level and does not change with the distribution of the non-zero coefficients of x
₀
. (iii) Despite the equality of the phase transition curves, different values of p show different performances once a small amount of measurement noise, w, is added.