On one-stage recovery for $\Sigma \Delta$-quantized compressed sensing

TL;DR

This paper extends one-stage reconstruction methods for $\Sigma \Delta$-quantized compressed sensing to broader classes of measurement matrices, including deterministic and structured random matrices, with theoretical guarantees.

Contribution

It introduces two new approaches for one-stage recovery in $\Sigma \Delta$-quantized CS applicable to more measurement matrices beyond sub-Gaussian types.

Findings

Extended theoretical guarantees to partial bounded orthonormal systems.

Included deterministic chirp sensing matrices in the analysis.

Demonstrated the effectiveness of the new methods with theoretical bounds.

Abstract

Compressed sensing (CS) is a signal acquisition paradigm to simultaneously acquire and reduce dimension of signals that admit sparse representations. When such a signal is acquired according to the principles of CS, the measurements still take on values in the continuum. In today's "digital" world, a subsequent quantization step, where these measurements are replaced with elements from a finite set is crucial. We focus on one of the approaches that yield efficient quantizers for CS: $\Sigma \Delta$ quantization, followed by a one-stage tractable reconstruction method, which was developed by Saab et al. with theoretical error guarantees in the case of sub-Gaussian matrices. We propose two alternative approaches that extend this result to a wider class of measurement matrices including (certain unitary transforms of) partial bounded orthonormal systems and deterministic constructions based on chirp sensing matrices.