The Whitney Reduction Network: Theory, Algorithms and Applications

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Kirby9973303 The investigator exploits the Whitney Reduction Network, a new tool for processing and analyzing data sets of high ambient dimension. Whitney's theorem motivates the architecture of the network, which consists of a "good projection," designed to be bilipschitz and retain the differential structure of the data, followed by a nonlinear inverse with good interpolation properties inherited from the choice of projection. The approach provides a natural representation of the data as the graph of a function that is approximated using radial basis functions (RBFs). The basic technique may be implemented in a variety of ways, hence the development and evaluation of a broad class algorithms is a primary focus of the research. Problems and algorithms examined include adaptive basis methods, RBF center selection, (spatio-temporal) clustering methods, direct and iterative methods for solving for RBF weights, RBF approximation with constraints and a local implementation of the basic algorithm. Scientific research has to a certain degree become a problem in data processing. High-speed numerical simulations of intractable model equations of natural phenomena produce enormous quantities of data. Understanding the phenomena means extracting and interpretting information in the data. The main goal of the project is to develop mathematical tools to produce efficient representations of data so as to expose the information it contains in a manner that the orignal data does not. Potential applications of this methodology are widespread and multi-disciplinary. In particular, the basic research undertaken here has direct applications through collaboration with Honeywell Corporation and the Air Force materials research program at Wright Patterson Air Force Base.