In investigating gaussian radial basis function (RBF) networks for their ability to model nonlinear time series, we have found that while RBF networks are much faster than standard sigmoid unit backpropagation for low-dimensional problems, their advantages diminish in high-dimensional input spaces. This is particularly troublesome if the input space contains irrelevant variables. We suggest that this limitation is due to the localized nature of RBFs. To gain the advantages of the highly nonlocal sigmoids and the speed advantages of RBFs, we propose a particular class of semilocal activation functions that is a natural interpolation between these two families. We present evidence that networks using these gaussian bar units avoid the slow learning problem of sigmoid unit networks, and, very importantly, are more accurate than RBF networks in the presence of irrelevant inputs. On the Mackey-Glass and Coupled Lattice Map problems, the speedup over sigmoid networks is so dramatic that the difference in training time between RBF and gaussian bar networks is minor. Gaussian bar architectures that superpose composed gaussians (gaussians-of-gaussians) to approximate the unknown function have the best performance. We postulate that an interesing behavior displayed by gaussian bar functions under gradient descent dynamics, which we call automatic connection pruning, is an important factor in the success of this representation.

A neural network with a single layer of hidden units of gaussian type is proved to be a universal approximator for real-valued maps defined on convex, compact sets of Rn.

We propose a simple architecture for implementing supervised neural network models optically with photorefractive technology. The architecture is very versatile: a wide range of supervised learning algorithms can be implemented including mean-field-theory, backpropagation, and Kanerva-style networks. Our architecture is based on a single crystal with spatial multiplexing rather than the more commonly used angular multiplexing. It handles hidden units and places no restrictions on connectivity. Associated with spatial multiplexing are certain physical phenomena, rescattering and beam depletion, which tend to degrade the matrix multiplications. Detailed simulations including beam absorption and grating decay show that the supervised learning algorithms (slightly modified) compensate for these degradations.