  Group-sparsity is a common low-complexity signal model with widespread
application across various domains of science and engineering. The recovery of
such signal ensembles from compressive measurements has been extensively
studied in the literature under the assumption that measurement operators are
modeled as densely populated random matrices. In this paper, we turn our
attention to an acquisition model intended to ease the energy consumption of
sensing devices by splitting the measurements up into distinct signal blocks.
More precisely, we present uniform guarantees for group-sparse signal recovery
in the scenario where a number of sensors obtain independent partial signal
observations modeled by block diagonal measurement matrices. We establish a
group-sparse variant of the classical restricted isometry property for block
diagonal sensing matrices acting on group-sparse vectors, and provide
conditions under which subgaussian block diagonal random matrices satisfy this
group-RIP with high probability. Two different scenarios are considered in
particular. In the first scenario, we assume that each sensor is equipped with
an independently drawn measurement matrix. We later lift this requirement by
considering measurement matrices with constant block diagonal entries. In other
words, every sensor is equipped with a copy of the same prototype matrix. The
problem of establishing the group-RIP is cast into a form in which one needs to
establish the concentration behavior of the suprema of chaos processes which
involves estimating Talagrand's $\gamma_2$ functional. As a side effect of the
proof, we present an extension to Maurey's empirical method to provide new
bounds on the covering number of sets consisting of finite convex combinations
of possibly infinite sets.
