On the Performance of Sparse Recovery via L_p-minimization (0<=p <=1)

It is known that a high-dimensional sparse vector x* in Rⁿ can be recovered from low-dimensional measurements y= A^{m*n} x* (m<n) . In this paper, we investigate the recovering ability of lp-minimization (0<=p<=1) as p varies, where lp-minimization returns a vector with the least lp ``norm'' among all the vectors x satisfying Ax=y. Besides analyzing the performance of strong recovery where lp-minimization needs 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 fixed sign pattern. When m/n goes to 1, we provide sharp thresholds of the sparsity ratio 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 m/n<1, we provide bounds of sparsity ratio for strong recovery and weak recovery respectively below which lp-minimization succeeds with overwhelming probability. Our bound of strong recovery improves on the existing bounds when m/n is large. Regarding the recovery threshold, 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 has better sparse recovery ability than l1-minimization since it is closer to l0-minimization. We provide an intuitive explanation to our findings and use numerical examples to illustrate the theoretical predictions.