Mathematical Sciences: Topics in Projection Pursuit, Neural Networks and Pattern Recognition

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Problems in the interface of statistics, mathematics and computational science will be studied, focusing on Fourier approximation and ridge estimation of regression and density functions and, in a separate context, on filtered Dirichlet distributions for local smoothing. In particular, preference orders will be studied for the frequency weights in multiple Fourier expansions and for the connection weights in single hidden-layer neural networks. Properties will be derived and applications such as feature extraction and projection pursuit will be considered in detail. This research will aid in automated learning from large data sets such as those arising in robotics, medicine and artificial intelligence. One aspect of the work will develop sophisticated mathematical techniques for application to problems in statistics and electrical engineering. A second aspect of the work will develop statistical methods to utilize an expert's knowledge directly without restricting the form in which the expert must express this information.