Quantization using Compressive Sensing

TL;DR

This paper investigates the optimal compression of sparse signals via compressive sensing, establishing rate-distortion bounds for quantized reconstructions and introducing a new matrix property ensuring these bounds.

Contribution

It introduces a novel restricted isometry-like property and proves the existence of matrices satisfying it, advancing theoretical understanding of quantization in compressive sensing.

Findings

Rate-distortion optimality for quantized compressive sensing signals

Introduction of a new restricted isometry-like property

Existence proof for matrices satisfying this property

Abstract

The problem of compressing a real-valued sparse source using compressive sensing techniques is studied. The rate distortion optimality of a coding scheme in which compressively sensed signals are quantized and then reconstructed is established when the reconstruction is also required to be sparse. The result holds in general when the distortion constraint is on the expected $p$-norm of error between the source and the reconstruction. A new restricted isometry like property is introduced for this purpose and the existence of matrices that satisfy this property is shown.