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An iterative algorithm for sparse and constrained recovery with applications to divergence-free current reconstructions in magneto-encephalography

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  • Ignace Loris
  • Caroline Verhoeven

Abstract

We propose an iterative algorithm for the minimization of a ℓ 1 -norm penalized least squares functional, under additional linear constraints. The algorithm is fully explicit: it uses only matrix multiplications with the three matrices present in the problem (in the linear constraint, in the data misfit part and in the penalty term of the functional). None of the three matrices must be invertible. Convergence is proven in a finite-dimensional setting. We apply the algorithm to a synthetic problem in magneto-encephalography where it is used for the reconstruction of divergence-free current densities subject to a sparsity promoting penalty on the wavelet coefficients of the current densities. We discuss the effects of imposing zero divergence and of imposing joint sparsity (of the vector components of the current density) on the current density reconstruction. Copyright Springer Science+Business Media, LLC 2013

Suggested Citation

  • Ignace Loris & Caroline Verhoeven, 2013. "An iterative algorithm for sparse and constrained recovery with applications to divergence-free current reconstructions in magneto-encephalography," Computational Optimization and Applications, Springer, vol. 54(2), pages 399-416, March.
    • Handle: RePEc:spr:coopap:v:54:y:2013:i:2:p:399-416
    DOI: 10.1007/s10589-012-9482-y
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    References listed on IDEAS

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    1. Giannone, Domenico & De Mol, Christine & Daubechies, Ingrid & Brodie, Joshua, 2007. "Sparse and Stable Markowitz Portfolios," CEPR Discussion Papers 6474, C.E.P.R. Discussion Papers.
    2. Patrick L. Combettes & Jean-Christophe Pesquet, 2011. "Proximal Splitting Methods in Signal Processing," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 185-212, Springer.
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