The "Minimum Reconstruction Error" Choice of Regularization Parameters: Some More Efficient Methods and Their Application to Deconvolution Problems

For the simple problem of estimatinga vector x 0 from a noisy data vector y = Bx 0 +e where B is a known ill-conditioned mn matrix and e is an unknown`white noise' vector, a classical regularized solution, say x() where > 0 is the regularization parameter, can be satisfactory provided is well chosen. Standard data-based methods for choosing (like generalized cross-validation, or GCV) are known to give a good estimate of the value of which minimizes the prediction error jjBx()? Bx 0 jj 2. In this paper, we focus on the minimization of the estimation (or reconstruction) error jjx() ? x 0 jj 2. We give suucient conditions for the existence of two unbiased estimators of the expectation of the inner product hx 0 ;x()i. This provides two estimates of the which minimizes jjx() ? x 0 jj 2 (The rst one was proposed by Rice in \Function estimates", Contemporary Mathematics ser., RI:AMS, no. 59, pp.137-151,1986]). We compare these two estimators in the case of deconvolution problems. In theory, the second estimator has no longer the possiblyìnnnite' variance of the rst one; however both are likely to produce frequent dramatic undersmoothing. Then we propose a third class of estimators based on automatic stabilization procedures, which are much more eecient in many deconvolution problems. This new approach for choosing regularization parameters can signiicantly improve on GCV especially for`severely' ill-conditioned problems. This is easily shown by analysing a simple example and is connrmed by numerical simulations with diierent degrees of ill-conditioning. 1. Introduction. 1.1. Regularized least squares. Suppose that we have to estimate x 0 2 IR n from a vector y = (y 1 ; : : :; y m) T of noisy data satisfying