Rank Restricted Isometry Property Implies the Rank Robust Null Space Property in Compressed Sensing for Matrix Recovery

Abstract

Compressed sensing refers to the recovery of high-dimensional but low-complexity entities from a small number of measurements. Two canonical examples of compressed sensing are the recovery of high-dimensional but sparse vectors, and high-dimensional but low-rank matrices. There is considerable literature on sufficient conditions for achieving these.In vector recovery, the restricted isometry property (RIP) and the robust null space property (RNSP) are two of the most commonly used sufficient conditions. Until recently, they 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 that any result in vector recovery that can be proved using the RIP can also be proved using the RNSP.In matrix recovery, the analogous sufficient conditions are the rank restricted isometry property (RRIP), and the rank robust null space property (RRNSP). Until now no relationship was available between the two properties. In the present paper, we show that the RRIP implies the RRNSP. Thus, as in the case of vector recovery, any result that can be proven using the rank restricted isometry property can also be proven using the rank restricted null space property.