On the Performance of Sparse Recovery via L_pminimization (0<=p <=1)
Abstract
It is known that a highdimensional sparse vector x* in R^n can be recovered from lowdimensional measurements y= A^{m*n} x* (m<n) . In this paper, we investigate the recovering ability of l_pminimization (0<=p<=1) as p varies, where l_pminimization returns a vector with the least l_p ``norm'' among all the vectors x satisfying Ax=y. Besides analyzing the performance of strong recovery where l_pminimization needs to recover all the sparse vectors up to certain sparsity, we also for the first time analyze the performance of ``weak'' recovery of l_pminimization (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 l_pminimization. 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 l_1minimization. We also explicitly demonstrate that l_pminimization (p<1) can return a denser solution than l_1minimization. For any m/n<1, we provide bounds of sparsity ratio for strong recovery and weak recovery respectively below which l_pminimization succeeds with overwhelming probability. Our bound of strong recovery improves on the existing bounds when m/n is large. Regarding the recovery threshold, l_pminimization has a higher threshold with smaller p for strong recovery; the threshold is the same for all p for sectional recovery; and l_1minimization can outperform l_pminimization for weak recovery. These are in contrast to traditional wisdom that l_pminimization has better sparse recovery ability than l_1minimization since it is closer to l_0minimization. We provide an intuitive explanation to our findings and use numerical examples to illustrate the theoretical predictions.
 Publication:

arXiv eprints
 Pub Date:
 November 2010
 arXiv:
 arXiv:1011.5936
 Bibcode:
 2010arXiv1011.5936W
 Keywords:

 Computer Science  Information Theory