Convergence and Stability of a Class of Iteratively Re-weighted Least Squares Algorithms for Sparse Signal Recovery in the Presence of Noise
In this paper, we study the theoretical properties of a class of iteratively re-weighted least squares (IRLS) algorithms for sparse signal recovery in the presence of noise. We demonstrate a one-to-one correspondence between this class of algorithms and a class of Expectation-Maximization (EM) algor...
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| Vydané v: | IEEE transactions on signal processing Ročník 62; číslo 1; s. 183 - 195 |
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| Hlavní autori: | , , , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
United States
30.10.2013
|
| ISSN: | 1053-587X |
| On-line prístup: | Získať plný text |
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| Shrnutí: | In this paper, we study the theoretical properties of a class of iteratively re-weighted least squares (IRLS) algorithms for sparse signal recovery in the presence of noise. We demonstrate a one-to-one correspondence between this class of algorithms and a class of Expectation-Maximization (EM) algorithms for constrained maximum likelihood estimation under a Gaussian scale mixture (GSM) distribution. The IRLS algorithms we consider are parametrized by 0 <
≤ 1 and
> 0. The EM formalism, as well as the connection to GSMs, allow us to establish that the IRLS(
,
) algorithms minimize
-smooth versions of the ℓ
'norms'. We leverage EM theory to show that, for each 0 <
≤ 1, the limit points of the sequence of IRLS(
,
) iterates are stationary point of the
-smooth ℓ
'norm' minimization problem on the constraint set. Finally, we employ techniques from Compressive sampling (CS) theory to show that the class of IRLS(
,
) algorithms is stable for each 0 <
≤ 1, if the limit point of the iterates coincides the global minimizer. For the case
= 1, we show that the algorithm converges exponentially fast to a neighborhood of the stationary point, and outline its generalization to super-exponential convergence for
< 1. We demonstrate our claims 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. |
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| Bibliografia: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 |
| ISSN: | 1053-587X |
| DOI: | 10.1109/TSP.2013.2287685 |