Reliable Recovery of Hierarchically Sparse Signals for Gaussian and Kronecker Product Measurements
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 (<inline-formula><tex-math notation="LaTeX">$\mathsf{HTP}$</tex-math></inline-formula>) algorithm referred to as <inline-formula><tex-math notation="LaTeX">$\mathsf{HiHTP}$</tex-math></inline-formula>. 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 <inline-formula><tex-math notation="LaTeX">$\mathsf{HTP}$</tex-math></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">$\mathsf{HiHTP}$</tex-math></inline-formula> 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, <inline-formula><tex-math notation="LaTeX">$\mathsf{HiHTP}$</tex-math></inline-formula> performs significantly better compared to <inline-formula><tex-math notation="LaTeX">$\mathsf{HTP}$</tex-math></inline-formula>.