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Journal of Fourier Analysis and Applications

, Volume 14, Issue 5–6, pp 629–654 | Cite as

Iterative Thresholding for Sparse Approximations

  • Thomas BlumensathEmail author
  • Mike E. Davies
Article

Abstract

Sparse signal expansions represent or approximate a signal using a small number of elements from a large collection of elementary waveforms. Finding the optimal sparse expansion is known to be NP hard in general and non-optimal strategies such as Matching Pursuit, Orthogonal Matching Pursuit, Basis Pursuit and Basis Pursuit De-noising are often called upon. These methods show good performance in practical situations, however, they do not operate on the 0 penalised cost functions that are often at the heart of the problem. In this paper we study two iterative algorithms that are minimising the cost functions of interest. Furthermore, each iteration of these strategies has computational complexity similar to a Matching Pursuit iteration, making the methods applicable to many real world problems. However, the optimisation problem is non-convex and the strategies are only guaranteed to find local solutions, so good initialisation becomes paramount. We here study two approaches. The first approach uses the proposed algorithms to refine the solutions found with other methods, replacing the typically used conjugate gradient solver. The second strategy adapts the algorithms and we show on one example that this adaptation can be used to achieve results that lie between those obtained with Matching Pursuit and those found with Orthogonal Matching Pursuit, while retaining the computational complexity of the Matching Pursuit algorithm.

Keywords

Sparse approximations Iterative thresholding 0 regularisation Subset selection 

Mathematics Subject Classification (2000)

15A29 41A46 68W25 68W40 90C27 

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Copyright information

© Birkhäuser Boston 2008

Authors and Affiliations

  1. 1.IDCOM & Joint Research Institute for Signal and Image ProcessingThe University of EdinburghEdinburghUK

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