Reliable recovery of hierarchically sparse signals for Gaussian and Kronecker product measurements

TL;DR

This paper introduces HiHTP, a new algorithm for reliably recovering hierarchically sparse signals from Gaussian and Kronecker product measurements, with improved theoretical guarantees and practical performance.

Contribution

The paper presents HiHTP, a novel variant of Hard Thresholding Pursuit, with proven convergence and enhanced sampling complexity bounds for hierarchically sparse signal recovery.

Findings

HiHTP outperforms HTP in numerical experiments.

Recovery guarantees are improved for noisy Gaussian measurements.

Efficient reconstruction from Kronecker product measurements is demonstrated.

Abstract

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.