On the Scaling Law for Compressive Sensing and its Applications

1 minimization can be used to recover sufficiently sparse unknown signals from compressed linear measurements. In fact, exact thresholds on the sparsity, as a function of the ratio between the system dimensions, so that with high probability almost all sparse signals can be recovered from i.i.d. Gaussian measurements, have been computed and are referred to as "weak thresholds" [4]. It has also been known that there is a tradeoff between the sparsity and the 1 minimization recovery stability in terms of the signal vector tail component. 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 (0 < < 1) from the weak threshold of 1 recovery, the parameter for the recovery stability will scale as 1 1. Our result is based on a careful analysis through the Grassmann angle framework for the Gaussian measurement matrix. We...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Weiyu Xu, Ao Tang
• Keywords: CORR 2010, Education, Minimization Algorithm, Recovery Stability, Weak Threshold