Compressed Sensing: How sharp is the Restricted Isometry Property

Compressed Sensing (CS) seeks to recover an unknown vector with $N$ entries by making far fewer than $N$ measurements; it posits that the number of compressed sensing measurements should be comparable to the information content of the vector, not simply $N$. CS combines the important task of compression directly with the measurement task. Since its introduction in 2004 there have been hundreds of manuscripts on CS, a large fraction of which develop algorithms to recover a signal from its compressed measurements. Because of the paradoxical nature of CS -- exact reconstruction from seemingly undersampled measurements -- it is crucial for acceptance of an algorithm that rigorous analyses verify the degree of undersampling the algorithm permits. The Restricted Isometry Property (RIP) has become the dominant tool used for the analysis in such cases. We present here an asymmetric form of RIP which gives tighter bounds than the usual symmetric one. We give the best known bounds on the RIP constants for matrices from the Gaussian ensemble. Our derivations illustrate the way in which the combinatorial nature of CS is controlled. Our quantitative bounds on the RIP allow precise statements as to how aggressively a signal can be undersampled, the essential question for practitioners. We also document the extent to which RIP gives precise information about the true performance limits of CS, by comparing with approaches from high-dimensional geometry.