Thresholding neural network for adaptive noise reduction

In the paper, a type of thresholding neural network (TNN) is developed for adaptive noise reduction. New types of soft and hard thresholding functions are created to serve as the activation function of the TNN. Unlike the standard thresholding functions, the new thresholding functions are infinitely differentiable. By using the new thresholding functions, some gradient-based learning algorithms become possible or more effective. The optimal solution of the TNN in a mean square error (MSE) sense is discussed. It is proved that there is at most one optimal solution for the soft-thresholding TNN. General optimal performances of both soft and hard thresholding TNNs are analyzed and compared to the linear noise reduction method. Gradient-based adaptive learning algorithms are presented to seek the optimal solution for noise reduction. The algorithms include supervised and unsupervised batch learning as well as supervised and unsupervised stochastic learning. It is indicated that the TNN with the stochastic learning algorithms can be used as a novel nonlinear adaptive filter. It is proved that the stochastic learning algorithm is convergent in certain statistical sense in ideal conditions. Numerical results show that the TNN is very effective in finding the optimal solutions of thresholding methods in an MSE sense and usually outperforms other noise reduction methods. Especially, it is shown that the TNN-based nonlinear adaptive filtering outperforms the conventional linear adaptive filtering in both optimal solution and learning performance.