  We propose and analyze a solution to the problem of recovering a block sparse
signal with sparse blocks from linear measurements. Such problems naturally
emerge inter alia in the context of mobile communication, in order to meet the
scalability and low complexity requirements of massive antenna systems and
massive machine-type communication. We introduce a new variant of the Hard
Thresholding Pursuit (HTP) algorithm referred to as HiHTP. We provide both a
proof of convergence and a recovery guarantee for noisy Gaussian measurements
that exhibit an improved asymptotic scaling in terms of the sampling complexity
in comparison with the usual HTP algorithm. Furthermore, hierarchically sparse
signals and Kronecker product structured measurements naturally arise together
in a variety of applications. We establish the efficient reconstruction of
hierarchically sparse signals from Kronecker product measurements using the
HiHTP algorithm. Additionally, we provide analytical results that connect our
recovery conditions to generalized coherence measures. Again, our recovery
results exhibit substantial improvement in the asymptotic sampling complexity
scaling over the standard setting. Finally, we validate in numerical
experiments that for hierarchically sparse signals, HiHTP performs
significantly better compared to HTP.
