Hierarchical restricted isometry property for Kronecker product measurements

Hierarchically sparse signals and Kronecker product structured measurements arise naturally in a variety of applications. The simplest example of a hierarchical sparsity structure is two-level $(s,\sigma)$-hierarchical sparsity which features $s$-block-sparse signals with $\sigma$-sparse blocks. For a large class of algorithms recovery guarantees can be derived based on the restricted isometry property (RIP) of the measurement matrix and model-based variants thereof. We show that given two matrices $\mathbf{A}$ and $\mathbf{B}$ having the standard $s$-sparse and $\sigma$-sparse RIP their Kronecker product $\mathbf{A}\otimes\mathbf{B}$ has two-level $(s,\sigma)$-hierarchically sparse RIP (HiRIP). This result can be recursively generalized to signals with multiple hierarchical sparsity levels and measurements with multiple Kronecker product factors. As a corollary we establish the efficient reconstruction of hierarchical sparse signals from Kronecker product measurements using the HiHTP algorithm. We argue that Kronecker product measurement matrices allow to design large practical compressed sensing systems that are deterministically certified to reliably recover signals in a stable fashion. We elaborate on their motivation from the perspective of applications.