Optimization of regularization parameter for sparse reconstruction based on predictive risk estimate

Recently, ℓ₁-based sparse reconstruction has shown its superior performance in terms of accuracy and interpretation. The reconstruction quality is generally sensitive to the value of regularization parameter. In this work, we develop two data-driven optimization schemes based on minimization of predicted Stein's unbiased risk estimate (p-SURE)-statistically equivalent to predicted mean squared error (p-MSE). First, we propose a recursive evaluation of p-SURE for any given regularization parameter, where the optimal value is identified by the minimum p-SURE. Second, for fast optimization, we perform alternating optimization between regularization parameter and solution during the ℓ₁-based iterative algorithms. We exemplify the proposed methods with a typical alternating direction method of multipliers (ADMM) for both analysis and synthesis formulations. Numerical experiments show that the proposed methods lead to highly accurate estimate of regularization parameter and nearly optimal reconstruction.