Optimal Modeling in Curved Reproducing Kernel Hilbert Spaces

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Principe AbstractThe objective of this research is to understand further the optimization of nonlinear systems with cost functions based on information theory. The recent interpretation of entropy as the mean square of the projected samples in feature space provides a link to the theory of reproducing kernels Hilbert spaces (RKHS), and raises the hypothesis that it may be possible to analytically compute the optimal solution of nonlinear systems, unlike the current estimates that use search procedures. The solution of the two most widely used models in filtering, the Wiener and the Kalman filters will be addressed. The approach is also novel because it will exploit both the inner product structure of the RKHS and the geometry of the estimation process using a differential geometry approach. Intellectual Merits. The intellectual merit of the proposal is to propose a new methodology based on information geometry to adapt systems with cost functions that directly manipulate information in the data, with the expected outcome of improving performance over the conventional methods in creating models from data and providing understanding of data. Broader Benefits. Our technology driven world is creating data at alarming rates. However, humans are interested in information, not data, and this is creating a tremendous bottleneck in medicine, business and even in engineering. Adaptive systems are one of the most promising methods to create models from data and provide understanding. The PI will also educate a breed of graduate students in the new area of differential geometry applied to signal processing who are needed to help solve this information bottleneck.