In this paper we show that discrete affine wavelet transforms can provide a tool for the analysis and synthesis of standard feedforward neural networks. Itis shown that wavelet frames for L2(IR) can be constructed based upon sigmoids. The spatia-spectral localization property of wavelets can be exploited in defining the topology and determining the weights of a feedforward network. Training a network constructed using the synthesis proceduredescribed here involves minimization of a convex cost functional andtherefore avoids pitfalls inherent in standard backpropagation algorithms. Extension of these methods to L2(IRN) is also discussed. 1 INTRODUCTION Feedforward type neural network models constructed from empirical data have been found to display significant predictive power [6]. Mathematical justification in support ofsuch predictive power may be drawn from various density and approximation theorems [1, 2, 5].

Bell-shaped response curves are commonly found in biological neurons whenever a natural metric exist on the corresponding relevant stimulus variable (orientation, position in space, frequency, time delay, ...). As a result, they are often used in neural models in different context ranging from resolution enhancement and interpolation tolearning (see, for instance, Baldi et al. (1988), Moody et al. (1989) *and Division of Biology, California Institute of Technology. The complete title of this paper should read: "Computing with arrays of bell-shaped and sigmoid functions.

We introduce oriented non-radial basis function networks (ONRBF) as a generalization of Radial Basis Function networks (RBF)- wherein the Euclidean distance metric in the exponent of the Gaussian is replaced bya more general polynomial. This permits the definition of more general regions and in particular-hyper-ellipses with orientations. Inthe case of hyper-surface estimation this scheme requires a smaller number of hidden units and alleviates the "curse of dimensionality" associatedkernel type approximators.In the case of an image, the hidden units correspond to features in the image and the parameters associated with each unit correspond to the rotation, scaling andtranslation properties of that particular "feature". In the context ofthe ONBF scheme, this means that an image can be represented by a small number of features. Since, transformation of an image by rotation, scaling and translation correspond to identical transformations of the individual features, the ONBF scheme can be used to considerable advantage for the purposes of image recognition and analysis.

F. Fallside We develop a sequential adaptation algorithm for radial basis function (RBF) neural networks of Gaussian nodes, based on the method of successive F-Projections.This method makes use of each observation efficiently in that the network mapping function so obtained is consistent with that information and is also optimal in the least L

We have created a radial basis function network that allocates a new computational unit whenever an unusual pattern is presented to the network. The network learns by allocating new units and adjusting the parameters of existing units. If the network performs poorly on a presented pattern, then a new unit is allocated which memorizes the response to the presented pattern. If the network performs well on a presented pattern, then the network parameters are updated using standard LMS gradient descent. For predicting the Mackey Glass chaotic time series, our network learns much faster than do those using back-propagation and uses a comparable number of synapses.

Function approximation on high-dimensional spaces is often thwarted by a lack of sufficient data to adequately "fill" the space, or lack of sufficient computational resources. The technique of local variable selection provides a partial solution to these problems by attempting to approximate functions locally using fewer than the complete set of input dimensions.

A new class of data structures called "bumptrees" is described. These structures are useful for efficiently implementing a number of neural network related operations. An empirical comparison with radial basis functions is presented on a robot ann mapping learning task.

A network based on splines is described.

Empirically, generalization between a training and a test stimulus falls off in close approximation to an exponential decay function of distance between the two stimuli in the "stimulus space" obtained by multidimensional scaling. Mathematically, thisresult is derivable from the assumption that an individual takes the training stimulus to belong to a "consequential" region that includes that stimulus but is otherwise of unknown location, size, and shape in the stimulus space (Shepard, 1987). As the individual gains additional information about the consequential region-by finding other stimuli to be consequential or nOl-the theory predicts the shape of the generalization function to change toward the function relating actual probability of the consequence to location in the stimulus space. This paper describes a natural connectionist implementation of the theory, and illustrates how implications of the theory for generalization, discrimination, and classification learning can be explored by connectionist simulation. 1 THE THEORY OF GENERALIZATION Because we never confront exactly the same situation twice, anything we have learned in any previous situation can guide us in deciding which action to take in the present situation only to the extent that the similarity between the two situations is sufficient to justify generalization of our previous learning to the present situation. Accordingly, principles of generalization must be foundational for any theory of behavior. In Shepard (1987) nonarbitrary principles of generalization were sought that would be optimum in any world in which an object, however distinct from other objects, is generally a member of some class or natural kind sharing some dispositional property of potential consequence for the individual. A newly encountered plant or animal might be edible or 665 666 Shepard and Kannappan poisonous.

Spherical Units can be used to construct dynamic reconfigurable consequential regions, the geometric bases for Shepard's (1987) theory of stimulus generalization in animals and humans. We derive from Shepard's (1987) generalization theory a particular multi-layer network with dynamic (centers and radii) spherical regions which possesses a specific mass function (Cauchy). This learning model generalizes the configural-cue network model (Gluck & Bower 1988): (1) configural cues can be learned and do not require pre-wiring the power-set of cues, (2) Consequential regions are continuous rather than discrete and (3) Competition amoungst receptive fields is shown to be increased by the global extent of a particular mass function (Cauchy). We compare other common mass functions (Gaussian; used in models of Moody & Darken; 1989, Krushke, 1990) or just standard backpropogation networks with hyperplane/logistic hidden units showing that neither fare as well as models of human generalization and learning.