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 ($\mathsf{HTP}$) algorithm referred to as $\mathsf{HiHTP}$. 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 $\mathsf{HTP}$ 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 $\mathsf{HiHTP}$ 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, $\mathsf{HiHTP}$ performs significantly better compared to $\mathsf{HTP}$.

中文翻译：

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可靠恢复高斯和克罗内克积测量的分层稀疏信号

我们提出并分析了从线性测量中恢复具有稀疏块的块稀疏信号问题的解决方案。这些问题自然会出现在移动通信的环境中，以满足海量天线系统和海量机器类型通信的可扩展性和低复杂度要求。我们引入了硬阈值追踪的新变体（$\mathsf{HTP}$) 算法称为 $\mathsf{HiHTP}$. 我们为噪声高斯测量提供收敛证明和恢复保证，与通常的相比，在采样复杂度方面表现出改进的渐近缩放$\mathsf{HTP}$算法。此外，分层稀疏信号和 Kronecker 积结构测量自然会在各种应用中一起出现。我们使用 Kronecker 乘积测量建立了分层稀疏信号的有效重建$\mathsf{HiHTP}$算法。此外，我们提供了将我们的恢复条件与广义相干性措施联系起来的分析结果。同样，我们的恢复结果在渐近采样复杂度缩放方面比标准设置有显着改善。最后，我们在数值实验中验证了对于分层稀疏信号，$\mathsf{HiHTP}$ 性能明显优于 $\mathsf{HTP}$.