On the performance of sparse recovery Via ℓp-minimization (0 ≤ p ≤ 1)

Abstract

It is known that a high-dimensional sparse vector x* in Rn can be recovered from low-dimensional measurements Ax * where Am×n(m < n) is the measurement matrix. In this paper, with A being a random Gaussian matrix, we investigate the recovering ability of ℓp-minimization (0≤ p ≤ 1) as p varies, where ℓp-minimization returns a vector with the least ℓp quasi-norm among all the vectors Ax = y. Besides analyzing the performance of strong recovery where ℓp-minimization is required to recover all the sparse vectors up to certain sparsity, we also for the first time analyze the performance of weak recovery of ℓp-minimization (0≤ p < 1) where the aim is to recover all the sparse vectors on one support with a fixed sign pattern. When α (:= m/n) → 1, we provide sharp thresholds of the sparsity ratio (i.e., percentage of nonzero entries of a vector) that differentiates the success and failure via ℓp- minimization. For strong recovery, the threshold strictly decreases from 0.5 to 0.239 as p increases from 0 to 1. Surprisingly, for weak recovery, the threshold is 2/3 for all p in [0,1), while the threshold is 1 for ℓ1- minimization. We also explicitly demonstrate that ℓp-minimization (p < 1) can return a denser solution than ℓ1-minimization. For any α ε (0,1), we provide bounds of the sparsity ratio for strong recovery and weak recovery, respectively, below which ℓp- minimization succeeds. Our bound of strong recovery improves on the existing bounds when α is large. In particular, regarding the recovery threshold, this paper argues that ℓp-minimization has a higher threshold with smaller p for strong recovery; the threshold is the same for all p for sectional recovery; and ℓ1-minimization can outperform ℓp-minimization for weak recovery. These are in contrast to traditional wisdom that ℓp-minimization, though computationally more expensive, always has better sparse recovery ability than ℓ1-minimization since it is closer to ℓ0- minimization. Finally, we provide an intuitive explanation to our findings. Numerical examples are also used to unambiguously confirm and illustrate the theoretical predictions. © 2011 IEEE.