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Group-sparsity is a common low-complexity signal model with widespreadapplication across various domains of science and engineering. The recovery ofsuch signal ensembles from compressive measurements has been extensivelystudied in the literature under the assumption that measurement operators aremodeled as densely populated random matrices. In this paper, we turn ourattention to an acquisition model intended to ease the energy consumption ofsensing devices by splitting the measurements up into distinct signal blocks.More precisely, we present uniform guarantees for group-sparse signal recoveryin the scenario where a number of sensors obtain independent partial signalobservations modeled by block diagonal measurement matrices. We establish agroup-sparse variant of the classical restricted isometry property for blockdiagonal sensing matrices acting on group-sparse vectors, and provideconditions under which subgaussian block diagonal random matrices satisfy thisgroup-RIP with high probability. Two different scenarios are considered inparticular. In the first scenario, we assume that each sensor is equipped withan independently drawn measurement matrix. We later lift this requirement byconsidering measurement matrices with constant block diagonal entries. In otherwords, every sensor is equipped with a copy of the same prototype matrix. Theproblem of establishing the group-RIP is cast into a form in which one needs toestablish the concentration behavior of the suprema of chaos processes whichinvolves estimating Talagrand's $\gamma_2$ functional. As a side effect of theproof, we present an extension to Maurey's empirical method to provide newbounds on the covering number of sets consisting of finite convex combinationsof possibly infinite sets.

The Restricted Isometry Property of Block Diagonal Matrices for Group-Sparse Signal Recovery