Wavelets as basis functions for localized learning in a multi-resolution hierarchy

An artificial neural network with one hidden layer of nodes, whose basis functions are drawn from a family of orthonormal wavelets, is developed. Wavelet networks or wave-nets are based on firm theoretical foundations of functional analysis. The good localization characteristics of the basis functions, both in the input and frequency domains, allow hierarchical, multi-resolution learning of input-output maps from experimental data. Wave-nets allow explicit estimation of global and local prediction error-bounds, and thus lend themselves to a rigorous and transparent design of the network. Computational complexity arguments prove that the training and adaptation efficiency of wave-nets is at least an order of magnitude better than other networks. The mathematical framework for the development of wave-nets is presented and various aspects of their practical implementation are discussed. The problem of predicting a chaotic time-series is solved as an illustrative example