For Matrix Recovery, Rank Restricted Isometry Property and Robust Uniform Boundedness Property Imply Rank Robust Null Space Property

Compressed sensing refers to the recovery of high-dimensional but low-complexity objects from a small number of measurements. The recovery of sparse vectors and the recovery of low-rank matrices are the main applications of compressed sensing theory. In vector recovery, the restricted isometry property (RIP) and the robust null space property (RNSP) are the two widely used sufficient conditions for achieving compressed sensing. Until recently, RIP and RNSP were viewed as two separate sufficient conditions. However, in a recent paper [1], the present authors have shown that in fact the RIP implies the RNSP, thus establishing the fact that RNSP is a weaker sufficient condition than RIP.In matrix recovery, there are three different sufficient con¬ditions for achieving low-rank matrix reconstruction, namely; Rank Restricted Isometry Property (RRIP), Rank Robust Null Space Property (RRNSP), and Robust Uniform Boundedness Property (RUBP). In this paper, using the result of [1], it is shown that actually both RRIP and RUBP imply the RRNSP, so that RRNSP is the weakest sufficient condition for matrix recovery. In contrast with the situation for vector recovery, until now there are no deterministic methods for designing a measurement operator for matrix recovery. The present results open the door towards such a possibility. © 2020 AACC