Mathematical Sciences: RUI: Applications of Wavelet Analysis to Neural Networks

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The investigator studies the approximation capabilities of afeedforward neural network for approximating functions in different function classes in terms of the activation function used, the number of layers, and the number of neurons. Particular emphasis is placed on dimension-independent bounds and localized approximation by networks. In particular, the research establishes strong connections between the theory of wavelets and neural networks. Applications to the construction of universal time series predictors and pattern classifiers also are studied. The project connects the theory of wavelets with that of neural networks. Both of these theories have numerous applications in such areas as data compression, prediction of time series, target classification, pattern recognition, and signal processing. Both topics of study have therefore flourished during the recent past, for the most part, independently of each other. There do exist certain similarities between the two theories stemming from the fact that they both deal with mathematically similar processes of approximation. The investigator explores these similarities further. In particular, he studies the inherent capabilities and limitations of universal mapping networks and develops efficient training paradigms. A remarkable feature of the networks to be studied is the ease with which the same network can be trained and retrained to perform a variety of tasks using a theoretically guaranteed minimal number of neurons.