In this paper, we study the problem of recovering a group sparse vector from a small number of linear measurements. In the past the common approach has been to use various "group sparsity-inducing" norms such as the Group LASSO norm for this purpose. By using the theory of convex relaxations, we show that it is also possible to use ℓ 1 -norm minimization for group sparse recovery. We introduce a new concept called group robust null space property (GRNSP), and show that, under suitable conditions, a group version of the restricted isometry property (GRIP) implies the GRNSP, and thus leads to group sparse recovery. When all groups are of equal size, our bounds are less conservative than known bounds. Moreover, our results apply even to situations where where the groups have different sizes. When specialized to conventional sparsity, our bounds reduce to one of the well-known "best possible" conditions for sparse recovery. This relationship between GRNSP and GRIP is new even for conventional sparsity, and substantially streamlines the proofs of some known results. Using this relationship, we derive bounds on the ℓ p -norm of the residual error vector for all p ∈ [1, 2], and not just when p = 2. When the measurement matrix consists of random samples of a sub-Gaussian random variable, we present bounds on the number of measurements, which are less conservative than currently known bounds.
Compressed sensing refers to the recovery of highdimensional 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 conditions 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.
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