Convergence and Stability of Iteratively Re-weighted Least Squares Algorithms for Sparse Signal Recovery in the Presence of Noise
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Brown_Convergence and stability.pdf
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Author(s)
Babadi, Behtash
Brown, Emery N.
Ba, Demba E.
Purdon, Patrick Lee
Alternative Title
Convergence and Stability of Iteratively Re-weighted Least Squares Algorithms
Date Issued
October 2013
Journal
IEEE Transactions on Signal Processing
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Citation
Ba, Demba, Behtash Babadi, Patrick L. Purdon, and Emery N. Brown. “Convergence and Stability of Iteratively Re-Weighted Least Squares Algorithms.” IEEE Transactions on Signal Processing 62, no. 1 (n.d.): 183–195.
Version
Original manuscript
Abstract
In this paper, we study the theoretical properties of iteratively re-weighted least squares (IRLS) algorithms and their utility in sparse signal recovery in the presence of noise. We demonstrate a one-to-one correspondence between the IRLS algorithms and a class of Expectation-Maximization (EM) algorithms for constrained maximum likelihood estimation under a Gaussian scale mixture (GSM) distribution. The EM formalism, as well as the connection to GSMs, allow us to establish that the IRLS algorithms minimize smooth versions of the lν `norms', for . We leverage EM theory to show that the limit points of the sequence of IRLS iterates are stationary points of the smooth lν “norm” minimization problem on the constraint set. We employ techniques from Compressive Sampling (CS) theory to show that the IRLS algorithm is stable, if the limit point of the iterates coincides with the global minimizer. We further characterize the convergence rate of the IRLS algorithm, which implies global linear convergence for ν = 1 and local super-linear convergence for . We demonstrate our results via simulation experiments. The simplicity of IRLS, along with the theoretical guarantees provided in this contribution, make a compelling case for its adoption as a standard tool for sparse signal recovery.
MIT Department
Harvard University--MIT Division of Health Sciences and Technology
Massachusetts Institute of Technology. Department of Brain and Cognitive Sciences
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DOI of Published Version
https://doi.org/10.1109/TSP.2013.2287685
