A new and improved quantitative recovery analysis for iterative hard thresholding algorithms in compressed sensing

TL;DR

This paper introduces a new analysis method for iterative hard thresholding algorithms in compressed sensing, providing improved conditions for convergence and recovery guarantees, especially for Gaussian measurement matrices and noise.

Contribution

It offers a novel fixed point-based recovery analysis for IHT and N-IHT algorithms, improving quantitative bounds on sparsity and undersampling trade-offs.

Findings

Derived sufficient and necessary conditions for convergence and fixed points.

Established lower bounds on phase transitions for guaranteed recovery.

Demonstrated substantial quantitative improvements over previous results.

Abstract

We present a new recovery analysis for a standard compressed sensing algorithm, Iterative Hard Thresholding (IHT) (Blumensath and Davies, 2008), which considers the fixed points of the algorithm. In the context of arbitrary measurement matrices, we derive a sufficient condition for convergence of IHT to a fixed point and a necessary condition for the existence of fixed points. These conditions allow us to perform a sparse signal recovery analysis in the deterministic noiseless case by implying that the original sparse signal is the unique fixed point and limit point of IHT, and in the case of Gaussian measurement matrices and noise by generating a bound on the approximation error of the IHT limit as a multiple of the noise level. By generalizing the notion of fixed points, we extend our analysis to the variable stepsize Normalised IHT (N-IHT) (Blumensath and Davies, 2010). For both stepsize schemes, we obtain lower bounds on asymptotic phase transitions in a proportional-dimensional framework, quantifying the sparsity/undersampling trade-off for which recovery is guaranteed. Exploiting the reasonable average-case assumption that the underlying signal and measurement matrix are independent, comparison with previous results within this framework shows a substantial quantitative improvement.