A Hierarchical View of Structured Sparsity in Kronecker Compressive Sensing

TL;DR

This paper introduces a hierarchical framework for structured sparsity in Kronecker compressed sensing, enabling efficient recovery algorithms and analysis of measurement matrices with improved computational performance.

Contribution

It presents a novel hierarchical view of Kronecker sparsity models and a two-stage recovery algorithm tailored to these structures, along with RIP analysis.

Findings

The proposed algorithm achieves comparable recovery accuracy to state-of-the-art methods.

It significantly reduces computational runtime in sparse recovery tasks.

The framework effectively captures multiple levels of sparsity in Kronecker-structured signals.

Abstract

Kronecker compressed sensing refers to using Kronecker product matrices as sparsifying bases and measurement matrices in compressed sensing. This work focuses on the Kronecker compressed sensing problem, encompassing three sparsity structures: $(i)$ a standard sparsity model with arbitrarily positioned nonzero entries, $(ii)$ a hierarchical sparsity model where nonzero entries are concentrated in a few blocks, each with only a subset of nonzero entries, and $(iii)$ a Kronecker-supported sparsity model where the support vector is a Kronecker product of smaller vectors. We present a hierarchal view of Kronecker compressed sensing that explicitly reveals a multiple-level sparsity pattern. This framework allows us to utilize the Kronecker structure of dictionaries and design a two-stage sparse recovery algorithm for different sparsity models. Further, we analyze the restricted isometry property of Kronecker-structured matrices under different sparsity models. Simulations show that our algorithm offers comparable recovery performance to state-of-the-art methods while significantly reducing runtime.