Inversion of homogeneous operators using stabilized hard thresholding with unknown noise variance

When inverting linear homogeneous operators, it is necessary to use regularization methods, since observed data are usually noisy. For noise suppression, threshold processing of wavelet coefficients of the observed signal function is often used. Threshold processing has become a popular noise suppression tool due to its simplicity, computational efficiency, and ability to adapt to functions that have different degrees of regularity at different domains. The paper discusses the recently proposed stabilized hard thresholding method that eliminates the main drawbacks of soft and hard thresholding methods and studies statistical properties of this method. In the data model with an additive Gaussian noise with unknown variance, an unbiased estimate of the mean square risk is analyzed and it is shown that under certain conditions, this estimate is asymptotically normal and the variance of the limit distribution depends on the type of estimate of noise variance.