Noisy 1-Bit Compressed Sensing Embeddings Enjoy a Restricted Isometry Property

TL;DR

This paper studies the properties of noisy 1-bit compressed sensing embeddings, demonstrating they satisfy a Restricted Isometry Property (RIP) even with noise, using VC dimension arguments.

Contribution

It introduces a new analysis showing noisy 1-bit compressed sensing embeddings satisfy RIP, extending previous results to noisy scenarios using VC dimension techniques.

Findings

Noisy 1-bit embeddings satisfy RIP under certain conditions.

The analysis uses VC dimension of sparse hemispheres.

The approach extends to additive white noise before quantization.

Abstract

We investigate the sign-linear embeddings of 1-bit compressed sensing given by Gaussian measurements. One can give short arguments concerning a Restricted Isometry Property of such maps using Vapnik-Chervonenkis dimension of sparse hemispheres. This approach has a natural extension to the presence of additive white noise prior to quantization. Noisy one-bit mappings are shown to satisfy an RIP when the metric on the sphere is given by the noise.