Self-organizing feature maps like the Kohonen map (Kohonen, 1989, Ritter et al., 1990) not only provide a plausible explanation for the formation of maps in brains, e.g. in the visual system (Obermayer et al., 1990), but have also been applied to problems like vector quantization, or robot arm control (Martinetz et al., 1990). The underlying organizing principle is the preservation of neighborhood relations. For this principle to lead to a most useful map, the topological structure of the output space must roughly fit the structure of the input data. However, in technical 1141 1142 Bauer, Pawelzik, and Geisel applications this structure is often not a priory known. For this reason several attempts have been made to modify the Kohonen-algorithm such, that not only the weights, but also the output space topology itself is adapted during learning (Kangas et al., 1990, Martinetz et al., 1991). Our contribution is also concerned with optimal output space topologies, but we follow a different approach, which avoids a possibly complicated structure of the output space. First we describe a quantitative measure for the preservation of neighborhood relations in maps, the topographic product P. The topographic product had been invented under the name of" wavering product" in nonlinear dynamics in order to optimize the embeddings of chaotic attractors (Liebert et al., 1991).

The Probabilistic Neural Network (PNN) algorithm represents the likelihood function of a given class as the sum of identical, isotropic Gaussians. In practice, PNN is often an excellent pattern classifier, outperforming other classifiers including backpropagation.

The localized linear discriminant network (LLDN) has been designed to address classification problems containing relatively closely spaced data from different classes (encounter zones [1], the accuracy problem [2]). Locally trained hyperplane segments are an effective way to define the decision boundaries for these regions [3]. The LLD uses a modified perceptron training algorithm for effective discovery of separating hyperplane/sigmoid units within narrow boundaries. The basic unit of the network is the discriminant receptive field (DRF) which combines the LLD function with Gaussians representing the dispersion of the local training data with respect to the hyperplane. The DRF implements a local distance measure [4], and obtains the benefits of networks oflocalized units [5]. A constructive algorithm for the two-class case is described which incorporates DRF's into the hidden layer to solve local discrimination problems. The output unit produces a smoothed, piecewise linear decision boundary. Preliminary results indicate the ability of the LLDN to efficiently achieve separation when boundaries are narrow and complex, in cases where both the "standard" multilayer perceptron (MLP) and k-nearest neighbor (KNN) yield high error rates on training data. 1 The LLD Training Algorithm and DRF Generation The LLD is defined by the hyperplane normal vector V and its "midpoint" M (a translated origin [1] near the center of gravity of the training data in feature space).

We derive criteria for training adaptive classifier networks to perform unsupervised data analysis. The first criterion turns a simple Gaussian classifier into a simple Gaussian mixture analyser. The second criterion, which is much more generally applicable, is based on mutual information.

G/SPLINES is an algorithm for building functional models of data. It uses genetic search to discover combinations of basis functions which are then used to build a least-squares regression model. Because it produces a population of models which evolve over time rather than a single model, it allows analysis not possible with other regression-based approaches. 1 INTRODUCTION G/SPLINES is a hybrid of Friedman's Multivariable Adaptive Regression Splines (MARS) algorithm (Friedman, 1990) with Holland's Genetic Algorithm (Holland, 1975). G/SPLINES has advantages over MARS in that it requires fewer least-squares computations, is easily extendable to non-spline basis functions, may discover models inaccessible to local-variable selection algorithms, and allows significantly larger problems to be considered. These issues are discussed in (Rogers, 1991). This paper begins with a discussion of linear regression models, followed by a description of the G/SPLINES algorithm, and finishes with a series of experiments illustrating its performance, robustness, and analysis capabilities.

Automatic determination of proper neural network topology by trimming oversized networks is an important area of study, which has previously been addressed using a variety of techniques. In this paper, we present Information Measure Based Skeletonisation (IMBS), a new approach to this problem where superfluous hidden units are removed based on their information measure (1M). This measure, borrowed from decision tree induction techniques, reflects the degree to which the hyperplane formed by a hidden unit discriminates between training data classes. We show the results of applying IMBS to three classification tasks and demonstrate that it removes a substantial number of hidden units without significantly affecting network performance.

A constructive algorithm is proposed for feed-forward neural networks, which uses node-splitting in the hidden layers to build large networks from smaller ones. The small network forms an approximate model of a set of training data, and the split creates a larger more powerful network which is initialised with the approximate solution already found. The insufficiency of the smaller network in modelling the system which generated the data leads to oscillation in those hidden nodes whose weight vectors cover regions in the input space where more detail is required in the model. These nodes are identified and split in two using principal component analysis, allowing the new nodes t.o cover the two main modes of each oscillating vector. Nodes are selected for splitting using principal component analysis on the oscillating weight vectors, or by examining the Hessian matrix of second derivatives of the network error with respect to the weight.s.

Terence D. Sanger Richard S. Sutton Christopher J. Matheus Massachusetts Institute GTE Laboratories GTE Laboratories of Technology Incorporated Incorporated Room E25-534 40 Sylvan Road 40 Sylvan Road Cambridge, MA 02139 Waltham, MA 02254 Waltham, MA 02254 tds@ai.mit.edu Abstract We present an iterative algorithm for nonlinear regression based on construction of sparse polynomials. Polynomials are built sequentially from lower to higher order. Selection of new terms is accomplished using a novel look-ahead approach that predicts whether a variable contributes to the remaining error. The algorithm is based on the tree-growing heuristic in LMS Trees which we have extended to approximation of arbitrary polynomials of the input features.

This paper is concerned with the problem of learning in networks where some or all of the functions involved are not smooth. Examples of such networks are those whose neural transfer functions are piecewise-linear and those whose error function is defined in terms of the 100 norm. Up to now, networks whose neural transfer functions are piecewise-linear have received very little consideration in the literature, but the possibility of using an error function defined in terms of the 100 norm has received some attention. In this paper we draw upon some recent results from the field of nonsmooth optimization (NSO) to present an algorithm for the non smooth case. Our motivation for this work arose out of the fact that we have been able to show that, in backpropagation, an error function based upon the 100 norm overcomes the difficulties which can occur when using the 12 norm. 1 INTRODUCTION This paper is concerned with the problem of learning in networks where some or all of the functions involved are not smooth.

Feedforward networks composed of units which compute a sigmoidal function of a weighted sum of their inputs have been much investigated. We tested the approximation and estimation capabilities of networks using functions more complex than sigmoids. Three classes of functions were tested: polynomials, rational functions, and flexible Fourier series. Unlike sigmoids, these classes can fit nonmonotonic functions. They were compared on three problems: prediction of Boston housing prices, the sunspot count, and robot arm inverse dynamics. The complex units attained clearly superior performance on the robot arm problem, which is a highly nonmonotonic, pure approximation problem. On the noisy and only mildly nonlinear Boston housing and sunspot problems, differences among the complex units were revealed; polynomials did poorly, whereas rationals and flexible Fourier series were comparable to sigmoids. 1 Introduction