The bi-sparse blind deconvolution problem is studied -- that is, from the knowledge of $h*(Qb)$, where $Q$ is some linear operator, recovering $h$ and $b$, which are both assumed to be sparse. The approach rests upon lifting the problem to a linear one, and then applying the hierarchical sparsity framework. In particular, the efficient HiHTP algorithm is proposed for performing the recovery. Then, under a random model on the 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：研究了双稀疏盲反卷积问题——即从$h*(Qb)$（其中$Q$为某线性算子）的已知信息中，恢复均假设为稀疏的$h$和$b$。该方法基于将问题提升至线性形式，并应用分层稀疏性框架。具体而言，提出了高效的HiHTP算法以实现恢复过程。随后，在矩阵$Q$的随机模型下，从理论上证明了：当满足条件$\mu \succcurlyeq s\log(s)^2\log(\mu)\log(\mu n) + s\sigma \log(n)$时，能以高概率恢复$s$-稀疏的$h \in \mathbb{K}^\mu$和$\sigma$-稀疏的$b \in \mathbb{K}^n$。