Title: Hierarchical Multi-Stage Recovery Framework for Kronecker Compressed Sensing
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
Keywords: Compressed sensing;Kronecker product;Restricted isometry property;Hierarchical sparsity;Tensor operation

Abstract: 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.

TL;DR: We study the Kronecker compressed sensing where we demonstrate a hierarchical view of the Kronecker compressed sensing and develop a versatile algorithmic and theoretical framework.