A recursive predictive risk estimate for proximal algorithms

Abstract

For accurate signal reconstruction, proximal gradient methods generally require proper selection of regularization parameter. In this paper, we develop two data-driven optimization schemes, based on minimization of unbiased predictive risk estimate (UPRE). First, we propose a recursive UPRE to estimate the prediction error during the proximal iterations, which can be used to optimize the regularization parameter. Second, for fast optimization, we parametrize each proximal iterate as a linear combination of few elementary functions (LET), and solve the linear weights by minimizing recursive UPRE. We further exemplify the proposed approaches with the basic iterative shrinkage/thresholding (IST) algorithms for ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -minimization. Numerical experiments show that iterating this process leads to higher reconstruction accuracy with remarkably faster computational speed than standard IST.