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Iterative thresholding for sparse approximations

Record type: Article

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.

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Citation

Blumensath, Thomas and Davies, Mike E. (2008) Iterative thresholding for sparse approximations The Journal of Fourier Analysis and Applications, 14, (5-6), pp. 629-654. (doi:10.1007/s00041-008-9035-z).

More information

Published date: December 2008
Keywords: sparse approximations, iterative thresholding, ? 0 regularisation, subset selection
Organisations: Other, Signal Processing & Control Grp

Identifiers

Local EPrints ID: 142521
URI: http://eprints.soton.ac.uk/id/eprint/142521
PURE UUID: 04dbe029-34e3-4ffd-9a1e-4cbbd022d09a

Catalogue record

Date deposited: 31 Mar 2010 15:09
Last modified: 18 Jul 2017 23:11

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