General Deviants: An Analysis of Perturbations in Compressed Sensing

TL;DR

This paper extends the analysis of compressed sensing recovery to include both additive noise and matrix perturbations, showing stability limits and confirming linear error growth through simulations.

Contribution

It introduces a comprehensive perturbation model for compressed sensing and analyzes the stability of Basis Pursuit recovery under these conditions.

Findings

Recovery stability is limited by observation noise level.

Reconstruction error is proportional to the perturbation magnitude.

Numerical simulations confirm linear error growth with perturbation.

Abstract

We analyze the Basis Pursuit recovery of signals with general perturbations. Previous studies have only considered partially perturbed observations Ax + e. Here, x is a signal which we wish to recover, A is a full-rank matrix with more columns than rows, and e is simple additive noise. Our model also incorporates perturbations E to the matrix A which result in multiplicative noise. This completely perturbed framework extends the prior work of Candes, Romberg and Tao on stable signal recovery from incomplete and inaccurate measurements. Our results show that, under suitable conditions, the stability of the recovered signal is limited by the noise level in the observation. Moreover, this accuracy is within a constant multiple of the best-case reconstruction using the technique of least squares. In the absence of additive noise numerical simulations essentially confirm that this error is a linear function of the relative perturbation.