Hierarchical Multi-Stage Recovery Framework for Kronecker Compressed Sensing

In this paper, we study the Kronecker compressed sensing problem, which focuses on recovering sparse vectors using linear measurements obtained using the Kronecker product of two or more matrices. We first introduce the hierarchical view of the Kronecker compressed sensing, showing that the Kronecker product measurement matrix probes the sparse vector from different levels, following a block-wise and hierarchical structure. Leveraging this insight, we develop a versatile multi-stage sparse recovery algorithmic framework and tailor it to three different sparsity models: standard, hierarchical, and Kronecker-supported. We further analyze the restricted isometry property of Kronecker product matrices under different sparsity models, and provide theoretical recovery guarantees for our multi-stage algorithm. Simulations demonstrate that our method achieves comparable recovery performance to other state-of-the-art techniques while substantially reducing run time owing to the hierarchical, multi-stage recovery process.