Sparse Reconstruction by Separable Approximation

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 ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">lscr</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ) error term added to a sparsity-inducing (usually lscr <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ) regularizater. We present an algorithmic framework for the more general problem of minimizing the sum of a smooth convex function and a nonsmooth, possibly nonconvex regularizer. We propose iterative methods in which each step is obtained by solving an optimization subproblem involving a quadratic term with diagonal Hessian (i.e., separable in the unknowns) plus the original sparsity-inducing regularizer; our approach is suitable for cases in which this subproblem can be solved much more rapidly than the original problem. Under mild conditions (namely convexity of the regularizer), we prove convergence of the proposed iterative algorithm to a minimum of the objective function. In addition to solving the standard lscr <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -lscr <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> case, our framework yields efficient solution techniques for other regularizers, such as an lscr <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> norm and group-separable regularizers. It also generalizes immediately to the case in which the data is complex rather than real. Experiments with CS problems show that our approach is competitive with the fastest known methods for the standard lscr <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -lscr <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> problem, as well as being efficient on problems with other separable regularization terms.