On the scaling law for compressive sensing and its applications

ℓ₁ minimization can be used to recover sufficiently sparse unknown signals from compressed linear measurements. In fact, exact thresholds on the sparsity (the size of the support set), under which with high probability a sparse signal can be recovered from i.i.d. Gaussian measurements, have been computed and are referred to as “weak thresholds” [4]. It was also known that there is a tradeoff between the sparsity and the ℓ₁ minimization recovery stability. In this paper, we give a closed-form characterization for this tradeoff which we call the scaling law for compressive sensing recovery stability. In a nutshell, we are able to show that as the sparsity backs off ω̅(0 <; ω̅ <; 1) from the weak threshold of ℓ₁ recovery, the parameter for the recovery stability will scale as 1/√(1-ω̅). Our result is based on a careful analysis through the Grassmann angle framework for the Gaussian measurement matrix. We will further discuss how this scaling law helps in analyzing the iterative reweighted ℓ₁ minimization algorithms. If the nonzero elements over the signal support follow a amplitude probability density function (pdf) f(·) whose t-th derivative f^t(0) ≠ 0 for some integer t ≥ 0, then a certain iterative reweighted ℓ₁ minimization algorithm can be analytically shown to lift the phase transition thresholds (weak thresholds) of the plain ℓ₁ minimization algorithm.