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Sparse reconstruction by separable approximation

Wright, S. ; Nowak, R. ; Figueiredo, M. A. T.

Sparse reconstruction by separable approximation, Proc IEEE International Conf. on Acoustics, Speech, and Signal Processing - ICASSP, Las Vegas, United States, Vol. , pp. - , April, 2008.

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Abstract
Finding sparse approximate solutions to large underdetermined linear systems of equations is a common problem in signal/image processing and statistics. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), wavelet-based deconvolution and reconstruction, and compressed sensing (CS) are a few well-known areas in which problems of this type appear. One standard approach is to minimize an objective function that includes a quadratic (l2) error term added to a sparsity-inducing (usually l1) regularizer. We present an algorithmic framework for the more general problem of minimizing the sum of a smooth convex function and a nonsmooth, possibly nonconvex, sparsity-inducing function. We propose iterative methods in which each step is an optimization subproblem involving a separable quadratic term (diagonal Hessian) plus the original sparsity-inducing term. Our approach is suitable for cases in which this subproblem can be solved much more rapidly than the original problem. In addition to solving the standard l2-l1 case, our approach handles other problems, e.g., lp regularizers with p < 1, or group-separable (GS) regularizers. Experiments with CS problems show that our approach provides state-of-the-art speed for the standard l2-l1 problem, and is also efficient on problems with GS regularizers.