  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.
