2016/06/19 by Shashank Ranjan, M. Vidyasagar, Ranjan, Shashank +1
Engineering · #FOS: Computer and information sciences #Machine Learning (stat.ML) #Microwave Imaging and Scattering Analysis #Photoacoustic and Ultrasonic Imaging #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1606.05889
openalex publication_date 2016/06/19 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
In this paper, we study the problem of recovering a group sparse vector from\na small number of linear measurements. In the past the common approach has been\nto use various "group sparsity-inducing" norms such as the Group LASSO norm for\nthis purpose. By using the theory of convex relaxations, we show that it is\nalso possible to use \ℓ1-norm minimization for group sparse recovery. We\nintroduce a new concept called group robust null space property (GRNSP), and\nshow that, under suitable conditions, a group version of the restricted\nisometry property (GRIP) implies the GRNSP, and thus leads to group sparse\nrecovery. When all groups are of equal size, our bounds are less conservative\nthan known bounds. Moreover, our results apply even to situations where where\nthe groups have different sizes. When specialized to conventional sparsity, our\nbounds reduce to one of the well-known "best possible" conditions for sparse\nrecovery. This relationship between GRNSP and GRIP is new even for conventional\nsparsity, and substantially streamlines the proofs of some known results. Using\nthis relationship, we derive bounds on the \ℓp-norm of the residual error\nvector for all p \∈ [1,2], and not just when p = 2. When the measurement\nmatrix consists of random samples of a sub-Gaussian random variable, we present\nbounds on the number of measurements, which are less conservative than\ncurrently known bounds.\n