Bisparse Blind Deconvolution through Hierarchical Sparse Recovery

Abstract

The hierarchical sparsity framework, and in particular the HiHTP algorithm, has been successfully applied to many relevant communication engineering problems recently, particularly when the signal space is hierarchically structured. In this paper, the applicability of the HiHTP algorithm for solving the bi-sparse blind deconvolution problem is studied. The bi-sparse blind deconvolution setting here consists of recovering $h$ and $b$ from the knowledge of $h\ast (Qb)$, where $Q$ is some linear operator, and both $b$ and $h$ are both assumed to be sparse. The approach rests upon lifting the problem to a linear one, and then applying HiHTP, through the \emph{hierarchical sparsity framework}. %In particular, the efficient HiHTP algorithm is proposed for performing the recovery. Then, for a Gaussian draw of the random matrix $Q$, it is theoretically shown that an $s$-sparse $h\in {\mathbb{K}}^{\hat{I}}¼$ and $\ddot{I}ƒ$-sparse $b\in {\mathbb{K}}^n$ with high probability can be recovered when $\hat{I}¼\succcurlyeq s\log (s{)}^2\log (\hat{I}¼)\log (\hat{I}¼n)+s\ddot{I}ƒ\log (n)$.

V3: Completely rewritten introduction, and a few corrections in the proof section. V2: Completely revised version, entirely different proof, resulting in the recovery guarantee improved by a factor s