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.
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Acknowledgements
I.L. is research associate of the Fonds de la recherche Scientifique-FNRS (Belgium). Part of this research was done while the authors were at the Computational and Applied Mathematics Programme of the Vrije Universiteit Brussel and was supported by VUB GOA-062 and by the Fonds voor Wetenschappelijk Onderzoek-Vlaanderen grant G.0564.09N. The authors would like to thank M. Fornasier and F. Pitolli for the useful discussions on MEG and the referees for their constructive comments.
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Loris, I., Verhoeven, C. An iterative algorithm for sparse and constrained recovery with applications to divergence-free current reconstructions in magneto-encephalography. Comput Optim Appl 54, 399–416 (2013). https://doi.org/10.1007/s10589-012-9482-y
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DOI: https://doi.org/10.1007/s10589-012-9482-y