Reliable Recovery of Hierarchically Sparse Signals for Gaussian and Kronecker Product Measurements
Journal article, 2020

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.

recovery guarantee

restricted isometry property

machine-type communications

Kronecker product

hard thresholding

block sparse

Inverse problem

compressed sensing

sparse vectors

hierarchical sparsity

channel estimation


pursuit algorithms


Ingo Roth

Freie Universität Berlin

Martin Kliesch

Heinrich Heine University Düsseldorf

Axel Flinth

Chalmers, Mathematical Sciences, Analysis and Probability Theory

Gerhard Wunder

Freie Universität Berlin

Jens Eisert

Freie Universität Berlin

IEEE Transactions on Signal Processing

1053-587X (ISSN)

Vol. 68 4002-4016 9120242

Subject Categories

Control Engineering

Signal Processing

Other Electrical Engineering, Electronic Engineering, Information Engineering



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