Title
A New and Improved Quantitative Recovery Analysis for Iterative Hard Thresholding Algorithms in Compressed Sensing
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 the 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 (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 tradeoff 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.
Year
DOI
Venue
2013
10.1109/TIT.2015.2399919
IEEE Transactions on Information Theory
Keywords
DocType
Volume
gaussian measurement matrix,improved quantitative recovery analysis,fixed points,algorithms,sparse signal recovery analysis,arbitrary measurement matrix,approximation theory,compressed sensing,gaussian distribution,average-case assumption,matrix algebra,convergence,iterative hard thresholding algorithms,proportional-dimensional framework,asymptotic phase transitions,variable stepsize normalised iht,approximation error,sparsity-undersampling tradeoff,iterative methods,probability,noise measurement,numerical analysis,vectors,noise,algorithm design and analysis
Journal
61
Issue
ISSN
Citations 
4
0018-9448
7
PageRank 
References 
Authors
0.52
17
2
Name
Order
Citations
PageRank
Coralia Cartis145128.74
Andrew Thompson2596.46