Robust recovery of signals with partially known support information using weighted BPDN

In this paper, some theoretical results are established for the weighted Basis Pursuit De-Noising (BPDN) to guarantee the robust signal recovery when partial support information of the signals is known in advance. To be specific, equipped with two popular and powerful restricted isometry property and coherence tools, we obtain two kinds of deterministic performance guarantees of this weighted BPDN, including some recovery conditions and their resultant error estimates. These results demonstrate that if half of the support information at least is assumed to be accurate, then the weighted BPDN will perform robust under much weaker conditions than the analogous ones for the BPDN. This theoretical finding not only coincides with the ones established previously for the constrained weighted l(1)-minimization model, but also well complements the previous investigation of the weighted BPDN that is based on the asymptotic analysis. Moreover, the extensive numerical experiments are also conducted to support the assessed performance of the weighted BPDN. Finally, we also discuss the potential to apply the obtained results to deal with the robust signal recovery when the multiple support information of signals becomes available.