Non-Convex Compressed Sensing Using Partial Support Information

In this paper we address the recovery conditions of weighted ${\mathrm{\ell }}_p$ minimization for signal reconstruction from compressed sensing measurements when partial support information is available. We show that weighted ${\mathrm{\ell }}_p$ minimization with $0<p<1$ is stable and robust under weaker sufficient conditions compared to weighted ${\mathrm{\ell }}_1$ minimization. Moreover, the sufficient recovery conditions of weighted ${\mathrm{\ell }}_p$ are weaker than those of regular ${\mathrm{\ell }}_p$ minimization if at least $50$ of the support estimate is accurate. We also review some algorithms which exist to solve the non-convex ${\mathrm{\ell }}_p$ problem and illustrate our results with numerical experiments.