ADA-PT: An Adaptive Parameter Tuning Strategy Based on the Weighted Stein Unbiased Risk Estimator

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The performance of iterative algorithms aimed at solving a regularized least squares problem typically depends on the value of some regularization parameter. Tuning the regularization parameter value is a fundamental step necessary to control the strength of the regularization and hence ensure a good performance. We address the problem of finding the optimal regularization parameter in such iterative algorithms. We propose to adaptively adjust the regularization parameter throughout the iterations of the algorithm by minimizing an estimate of the current risk, typically the Weighted Stein unbiased risk estimate (WSURE). We then prove that, for the case of the Tikhonov regularization, the proposed ADAptive Parameter Tuning (ADA-PT) strategy provides a stationary point consistent with the risk minimizer. We illustrate the efficiency of ADA-PT on two image deconvolution problems: one with the Tikhonov regularization and one with the weighted $\ell-1$ analysis wavelet regularization.