Hierarchical restricted isometry property for Kronecker product measurements

Roth I, Flinth A, Kueng R, Eisert J, Wunder G (2019)

Publication Type: Conference contribution

Publication year: 2019

Publisher: Institute of Electrical and Electronics Engineers Inc.

Pages Range: 632-638

Conference Proceedings Title: 2018 56th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2018

Event location: Monticello, IL, USA

ISBN: 9781538665961

DOI: 10.1109/ALLERTON.2018.8635829

Abstract

Hierarchically sparse signals and Kronecker product structured measurements naturally arise 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 s-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 A and 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.

Involved external institutions

Freie Universität Berlin DE Germany (DE) Université Toulouse III - Paul Sabatier FR France (FR) California Institute of Technology (Caltech) US United States (USA) (US)

How to cite

APA:

Roth, I., Flinth, A., Kueng, R., Eisert, J., & Wunder, G. (2019). Hierarchical restricted isometry property for Kronecker product measurements. In 2018 56th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2018 (pp. 632-638). Monticello, IL, USA: Institute of Electrical and Electronics Engineers Inc..

MLA:

Roth, Ingo, et al. "Hierarchical restricted isometry property for Kronecker product measurements." Proceedings of the 56th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2018, Monticello, IL, USA Institute of Electrical and Electronics Engineers Inc., 2019. 632-638.

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