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Two-point step-size iterative soft-thresholding method for sparse reconstruction

Many problems in signal processing involve minimizing a quadratic error term combined with an ℓ1 norm term. Iterative soft-thresholding (IST) algorithm is a basic method for these problems. Despite previous explanations of IST, this study presents it as a method of constructing a local model to approximate the objective function. It only uses the approximation of the quadratic term while keeping the ℓ1 norm term unchanged. Based on this, we propose a modified IST (MIST), using a general strictly convex quadratic function to approximate the quadratic part. IST uses the identity matrix to approximate the Hessian matrix of the quadratic term, while we adopt an adaptive matrix by using the information of current and former iterates. This strategy results in the two-point step-size IST, including MISTBB1 and MISTBB2. Various experiments on compressed sensing show that MISTBB1 is much faster than competing codes and insensitive to the regularization parameter.