# Networks for approximation and learning

Article Abstract:

Learning an input-output mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multi-dimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, this form of learning is closely related to classical approximation techniques, such as generalized splines and regularization theory. This paper considers the problem of the approximation of nonlinear mappings - especially continuous mappings. We develop a theoretical framework for approximation based on regularization techniques that leads to a class of three-layer networks that we call regularization networks and include as a special case the well-known Radial Basis Functions method. Regularization networks are not only equivalent to generalized splines, but are also closely related to pattern recognition methods such as Parzen windows and potential functions and to several neural network algorithms, such as Kanerva's associative memory, backpropagation, and Kohonen's topology preserving map. They also have an interesting interpretation in terms of prototypes that are synthesized and optimally combined during the learning stage. This paper generalizes the theory of regularization networks to a formulation that turns out to include task-dependent clustering and dimensionality reduction. We also discuss briefly some intriguing analogies with neurobiological data. (Reprinted by permission of the publisher.)

Publication Name: Proceedings of the IEEE

Subject: Electronics

ISSN: 0018-9219

Year: 1990

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# Fast frequency synthesis by PLL using a continuous phase divider

Article Abstract:

A method of constructing a fast settling time frequency synthesizer is possible using a double-loop PLL with a short convergence time and continuous phase diver which matches the VCO and reference frequencies to each other. The new FS also does not require a loop filter. The new method has been simulated on a digital computer with good results.

Publication Name: Proceedings of the IEEE

Subject: Electronics

ISSN: 0018-9219

Year: 1988

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# Chebyshev Phase Approximation

Article Abstract:

It is difficult to determine the transfer function of a network with a Chebyshev norm in an explicit form. A method to determine the transfer function with minimum phase shift is described. The method is based on the solution of the linear equation. Characteristics of phase v-s. normalized frequency is graphed.

Publication Name: Proceedings of the IEEE

Subject: Electronics

ISSN: 0018-9219

Year: 1984

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