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In this paper, we propose a novel algorithm for structured sparsity reconstruction. This algorithm is based on the iterative reweighted least squares (IRLS) framework, and accelerated by the preconditioned conjugate gradient method. The convergence rate of the proposed algorithm is almost the same as that of the traditional IRLS algorithms, that is, exponentially fast. Moreover, with the devised preconditioner, the computational cost for each iteration is significantly less than that of traditional IRLS algorithms, which makes it feasible for large scale problems. Besides the fast convergence, this algorithm can be flexibly applied to standard sparsity, group sparsity, and overlapping group sparsity problems. Experiments are conducted on a practical application compressive sensing magnetic resonance imaging. Results demonstrate that the proposed algorithm achieves superior performance over 9 state-of-the-art algorithms in terms of both accuracy and computational cost.

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