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.
Highlights
The general idea of compressed sensing [11] is to exploit sparsity of a signal x to facilitate its recovery from incomplete and noisy linear measurements
Hierarchical sparsity is a structured notion of sparsity that arises in many applications
The structure allows for efficient custom-tailored recovery algorithms, such as the hierarchical hard threshold pursuit (HTP) (HiHTP), but it is more restrictive yielding stronger guarantees than the ones given by classical compressed sensing
Summary
The general idea of compressed sensing [11] is to exploit sparsity of a signal x to facilitate its recovery from incomplete and noisy linear measurements. A subset of the authors of this work has introduced a general RIP-based recovery framework for hierarchically sparse vectors in Refs. Theorem 3.2, we show that the RIP assumption on the top-level matrix can be relaxed if the sub-level matrices are mutually incoherent in a specific notion This provides a more detailed picture of HiRIP arising from the properties of the constituent matrices and introduces considerable flexibility to derive HiRIP for specific instances of hierarchical measurement operators. Compared results for block sparse or level-sparse signals, the analysis of hierarchical measurement operator crucially relies on the interplay between sparsity assumption on different hierarchy levels.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
