  Can compression algorithms be employed for recovering signals from their
underdetermined set of linear measurements? Addressing this question is the
first step towards applying compression algorithms for compressed sensing (CS).
In this paper, we consider a family of compression algorithms $\mathcal{C}_r$,
parametrized by rate $r$, for a compact class of signals $\mathcal{Q} \subset
\mathds{R}^n$. The set of natural images and JPEG at different rates are
examples of $\mathcal{Q}$ and $\mathcal{C}_r$, respectively. We establish a
connection between the rate-distortion performance of $\mathcal{C}_r$, and the
number of linear measurements required for successful recovery in CS. We then
propose compressible signal pursuit (CSP) algorithm and prove that, with high
probability, it accurately and robustly recovers signals from an
underdetermined set of linear measurements. We also explore the performance of
CSP in the recovery of infinite dimensional signals.
