Compressed Sensing and ΣΔ-Quantization

by Joe-Mei Feng
Doctoral thesis
Date of Examination:2018-02-12
Date of issue:2019-02-11
Advisor:Prof. Dr. Felix Krahmer
Referee:Prof. Dr. Gerlind Plonka-Hoch
Referee:Dr. Sjoerd Dirksen
Referee:Prof. Dr. Anita Schöbel
Referee:Prof. Dr. Russell Luke
Referee:Dr. Timo Aspelmeier
Persistent Address: http://dx.doi.org/10.53846/goediss-7272

 

 

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Abstract

English

The main issue of my thesis is to bound the error while recovering signals from their compressed and quantized form. Especially my central contribution is that, together with my co-authors, we provide the first analysis of a non-trivial quantization scheme for compressed sensing measurements arising from structured measurements. Specifically, our analysis studies compressed sensing matrices consisting of rows selected at random, without replacement, from a circulant matrix generated by a random subgaussian vector. We quantize the measurements using stable, possibly one-bit, Sigma-Delta schemes, and use a reconstruction method based on convex optimization. We show that the part of the reconstruction error due to quantization decays polynomially in the number of measurements. This is in-line with analogous results on Sigma-Delta quantization associated with random Gaussian or subgaussian matrices, and significantly better than results associated with the widely assumed memoryless scalar quantization. Moreover, we prove that our approach is stable and robust; i.e., the reconstruction error degrades gracefully in the presence of non-quantization noise and when the underlying signal is not strictly sparse. The analysis relies on results concerning subgaussian chaos processes as well as a variation of McDiarmid's inequality. Also my co- author and I provide a new approach to estimating the error of re- construction from ΣΔ-quantized compressed sensing measurements. Our method is based on the restricted isometry property (RIP) of a certain projection of the measurement matrix. Our result yields simple proofs and a slight generalization of the best-known reconstruction error bounds for Gaussian and subgaussian measurement matrices.
Keywords: Compressed; Sensing; Quantization; ΣΔ-Quantization
 
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