Abstract

Increasingly models, mappings, systems and algorithms used for signal processing need to be nonlinear in order to meet performance specifications in communications, computing and control systems applications. Simple computational models have been developed, including neural networks, which can efficiently implement a variety of nonlinear mappings through appropriate choice of model parameters. However, the design of arbitrary nonlinear mappings using these models and measured data requires both understanding how realizable (finite) systems perform if optimized given finite data, and a method for computing globally optimal system parameters. In this thesis, we use constructive homotopy methods both to geometrically explore the mapping capabilities of finite neural networks, and to rigorously develop a robust method for computing optimal solutions to systems of nonlinear equations which, like neural network equations, have an unknown number of solutionsand may have solutions at infinity.

Citations