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 $h*(Qb)$, 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 $h \in \mathbb{K}^\mu$ and $\sigma$-sparse $b \in \mathbb{K}^n$ with high probability can be recovered when $\mu \succcurlyeq s\log(s)^2\log(\mu)\log(\mu n) + s\sigma \log(n)$.

翻译：分层稀疏框架，特别是HiHTP算法，近年来已成功应用于许多相关通信工程问题，尤其是在信号空间具有分层结构的情况下。本文研究了HiHTP算法在解决双稀疏盲反卷积问题中的适用性。此处的双稀疏盲反卷积设定旨在从已知的 $h*(Qb)$ 中恢复 $h$ 和 $b$，其中 $Q$ 为某线性算子，且假设 $b$ 与 $h$ 均为稀疏信号。该方法通过将问题提升为线性问题，并借助\emph{分层稀疏框架}应用HiHTP算法实现恢复。%特别地，本文提出了高效的HiHTP算法以执行恢复过程。随后，对于随机矩阵 $Q$ 的高斯分布，理论分析表明：当 $\mu \succcurlyeq s\log(s)^2\log(\mu)\log(\mu n) + s\sigma \log(n)$ 时，能以高概率恢复 $s$-稀疏的 $h \in \mathbb{K}^\mu$ 与 $\sigma$-稀疏的 $b \in \mathbb{K}^n$。