The Group Restricted Isometry Property for Subgaussian Block Diagonal Matrices

We address the problem of reconstructing group-sparse vectors from compressive measurements acquired via subgaussian block diagonal measurement operators. Such results can be obtained by establishing the so-called group restricted isometry property of the underlying measurement matrix. In particular, the problem is reduced to the task of bounding certain geometric objects associated with the suprema of a particular chaos process, which involves estimating Talagrand’s γ 2 -functional via Dudley’s metric entropy integral. As part of the proof, we generalize Maurey’s empirical method to provide new bounds on the covering number of sets consisting of finite convex combinations of compact sets.