Nonlinear regularization of the inversion of linear homogeneous operators using the block thresholding method

The methods of thresholding the coefficients of wavelet expansions have become a popular tool for regularization of inverse statistical problems due to their simplicity, computational efficiency, and the ability to adapt both to the type of operators and to the features of the function under study. This approach proved to be the most fruitful for inversion of linear homogeneous operators arising in some signal and image processing problems. The paper considers the block thresholding method in which the decomposition coefficients are processed in groups that allows taking into account information about neighboring coefficients. In a data model with an additive Gaussian noise, an unbiased estimate of the mean-square risk is analyzed and it is shown that under certain conditions, this estimate is strongly consistent and asymptotically normal. These properties allow constructing asymptotic confidence intervals for the theoretical mean-square risk of the method under consideration.