Wavelet networks can in some situations be an attractive alternative to other neural networks such as sigmoidal feed-forward and Radial Basis Function (RBF) networks. However, theoretical issues on the construction of multi-dimensional wavelet frames and corresponding learning algorithms have not been satisfactorily investigated. Our first objective is to unambiguously determinine the conditions under which multi-dimensional wavelet frame can be constructed and then to develop learning algorithms and verify them via simulations. The second objective is to investigate the use of wavelet networks in adaptive control and test their effectiveness.
We generalise the sufficient conditions for one dimensional wavelet frames given by Daubechies to the multi-dimensional case for both single-scaling dilations in all dimensions and independent dilations in each dimension. We then study constructive procedures that provide multi-dimensional frames satisfying these sufficiency conditions. Tensor product as well as Radial construction methods are considered. Wavelet frame decomposition is then used for function approximation in the manner of neural networks. A heuristic methodology similar to Platt's Resource Allocating Network (RAN) for sequentially learning the network is developed, which attempts to force near-orthogonality. The approximation capability of wavelet networks is used in the stable adaptive control of non-linear systems in canonical form.
Explicit sufficient conditions for multi-dimensional wavelet frames derived in our work corrects some earlier erroneous assertions by other researchers and unambiguously show the conditions under which a multi-dimensional tensor or radial frame can be constructed. Our simulations also show the utility of our sequential learning methodology in building more compact networks. Moreover, Adaptive Control using a full wavelet network results in a small reduction in the number of nodes over a similar Gaussian RBF network, but this reduction is not drastic enough to make wavelet networks in their complete form significantly better than complete RBF networks in high dimensional applications.
Our results show an improvement over published results with respect to network size, especially in areas like monitoring, process control, and identification. A significant aspect is that learning is sequential, and data are presented only once, making it attractive in on-line applications. With some refinements, these methods could be applied successfully in the above mentioned industries especially when the dimension is small, and in many other areas such as signal/image processing, pattern recognition, etc.
Feasibility of irregular sampling methods for wavelet network construction needs investigation. The sequential learning strategy should be studied in a statistical framework and refined for high dimensional cases. More general classes of non-linear systems for adaptive control could be considered. Statistical bounds on network size, data complexity, etc are also of interest.