Can compression algorithms be employed to the problem of recovering signals from
their underdetermined set of linear measurements? Addressing this question is
the first step towards applying compression algorithms to compressed sensing
(CS). In this work, we consider a family of compression algorithms for a compact
class of signals, and establish a connection between its rate-distortion
performance  and the number of linear measurement required for successful
recovery of CS. We then propose the compressible signal pursuit (CSP) algorithm
and prove that, with high probability, it accurately and robustly recovers
signals from an underdetermined set of linear measurements.