Hierarchical Isometry Properties of Hierarchical Measurements

TL;DR

This paper introduces hierarchical measurement operators with proven isometry properties, enabling efficient and robust recovery of structured signals, and demonstrates their effectiveness through theoretical bounds and practical communication applications.

Contribution

It develops a new class of hierarchical measurement operators with theoretical guarantees and applies them to communication scenarios using the scalable HiHTP algorithm.

Findings

Hierarchical measurements satisfy stable restricted isometry properties.

Theoretical bounds extend previous Kronecker-product measurement results.

Numerical experiments show effective sparse signal recovery and block detection.

Abstract

A new class of measurement operators, coined hierarchical measurement operators, and prove results guaranteeing the efficient, stable and robust recovery of hierarchically structured signals from such measurements. We derive bounds on their hierarchical restricted isometry properties based on the restricted isometry constants of their constituent matrices, generalizing and extending prior work on Kronecker-product measurements. As an exemplary application, we apply the theory to two communication scenarios. The fast and scalable HiHTP algorithm is shown to be suitable for solving these types of problems and its performance is evaluated numerically in terms of sparse signal recovery and block detection capability.