A Research On Wavelet Networks Related Issues

Artificial neural network is a nonlinear dynamical system constructed by a largenumber of interconnected simple and nonlinear basic unit---neurons. From amathematical point of view, the artificial neural network is a function approximator.Wavelet based function approximator is also a linear combination of a large number ofsimple and nonlinear basic unit-the dilation and translation of a wavelet function. Aswavelet has a good localized time-frequency characteristic, it performs quite well infunction approximation. As a result, wavelet neural network, the combination ofwavelet and neural network, also has the good property of neural network and wavelet.Thus, the wavelet network is widely applied in image processing and signal analysisand other areas.Based on wavelet theory and neural network, this thesis discuss two waveletnetworks releated issues:1. The improvement of learning algorithm of wavelet networks. The learningalgorithm of neural network is a hot issue, and also a difficult one. The originalback-propagation algorithm fails to take advantage of the characteristics of wavelet. Inthis thesis, an algorithm called multiscale is proposed for the training of waveletnetwork. This algorithm is inspired by the multigrid method and takes advantage of themultiscale characteristics of wavelet. Based on Mallat algorithm, we constructed aninterpolation operator and a restriction operator. During the procedure of training, theinterpolation operator and restriction operator is used to adjust network weights betweendifferent scales. Numerical experiments show that the algorithm do speed up theconvergence rate of wavelet network training.2. The sparse approximation problem with regularized wavelet network. Since theproblem of function approximation based on wavelet network is an ill-posed one,regularization condition of sparse constraint on coefficients can help stabilize solution.After carefully review the meanings of sparse, energy window is proposed to measurethe degree of sparse. Redundant dictionaries construed by wavelet and energy windowas regularization condition helps make the sparse approximation problem into an extremum problem of the Tikhonov function, which simplifies the sparse approximationproblem. Numerical experiments show the effect of the method and a comparisonbetween different regularization conditions and arguments are made.