Minimax estimates of the loss function based on integral error probabilities during threshold processing of wavelet coefficients

Noise reduction is one of the main tasks of signal processing. Wavelet transform-based methods for solving this problem have proven to be reliable and effective. Thresholding methods that use the idea of a sparse representation of a signal function in the space of wavelet coefficients have become especially popular. These methods use fast nonlinear algorithms that adapt to the local features of the signal being processed. The parameters of these algorithms are selected based on some quality criterion or minimization of a given loss function. Most often, the mean square risk is considered as a loss function. However, in some applications, minimizing the mean square risk does not always lead to satisfactory results. In the present paper, the authors consider the loss function based on the integral probabilities of errors in calculating the wavelet coefficients. For hard and soft thresholding methods, the boundaries for the optimal threshold values are calculated and the minimax order of the considered loss function in the class of Lipschitz-regular signals is estimated.