Since a neural network predictor inherently has an excessive number of parameters, reducing the prediction error is usually done by reducing variance. Methods for reducing neural network complexity can be viewed as a regularization technique to reduce this variance. Examples of such methods are Optimal Brain Damage (Le Cun et.

Smoothing regularizers for radial basis functions have been studied extensively, but no general smoothing regularizers for projective basis junctions (PBFs), such as the widely-used sigmoidal PBFs, have heretofore been proposed. We derive new classes of algebraically-simple mH'-order smoothing regularizers for networks of the form f(W, x)

The essential property of BBNs is summarized by the Markov condition, which asserts that each variable is independent of its non-descendants given its parents.

We address statistical classifier design given a mixed training set consisting of a small labelled feature set and a (generally larger) set of unlabelled features. This situation arises, e.g., for medical images, where although training features may be plentiful, expensive expertise is required to extract their class labels. We propose a classifier structure and learning algorithm that make effective use of unlabelled data to improve performance. The learning is based on maximization of the total data likelihood, i.e. over both the labelled and unlabelled data subsets. Two distinct EM learning algorithms are proposed, differing in the EM formalism applied for unlabelled data. The classifier, based on a joint probability model for features and labels, is a "mixture of experts" structure that is equivalent to the radial basis function (RBF) classifier, but unlike RBFs, is amenable to likelihood-based training. The scope of application for the new method is greatly extended by the observation that test data, or any new data to classify, is in fact additional, unlabelled data - thus, a combined learning/classification operation - much akin to what is done in image segmentation - can be invoked whenever there is new data to classify. Experiments with data sets from the UC Irvine database demonstrate that the new learning algorithms and structure achieve substantial performance gains over alternative approaches.

When combining a set of learned models to form an improved estimator, the issue of redundancy or multicollinearity in the set of models must be addressed. A progression of existing approaches and their limitations with respect to the redundancy is discussed. A new approach, PCR *, based on principal components regression is proposed to address these limitations. An evaluation of the new approach on a collection of domains reveals that: 1) PCR* was the most robust combination method as the redundancy of the learned models increased, 2) redundancy could be handled without eliminating any of the learned models, and 3) the principal components of the learned models provided a continuum of "regularized" weights from which PCR * could choose.

When triangulating a belief network we aim to obtain a junction tree of minimum state space. According to (Rose, 1970), searching for the optimal triangulation can be cast as a search over all the permutations of the graph's vertices. Our approach is to embed the discrete set of permutations in a convex continuous domain D. By suitably extending the cost function over D and solving the continous nonlinear optimization task we hope to obtain a good triangulation with respect to the aformentioned cost. This paper presents two ways of embedding the triangulation problem into continuous domain and shows that they perform well compared to the best known heuristic.

The classes in classification tasks often have a natural ordering, and the training and testing examples are often incomplete. We propose a nonlinear ordinal model for classification into ordered classes. Predictive, simulation-based approaches are used to learn from past and classify future incomplete examples. These techniques are illustrated by making prognoses for patients who have suffered severe head injuries.

Further details may be found in (Lowe, 1993; Lowe and Tipping, 1996). We seek a dimension-reducing, topographic transformation of data for the purposes of visualisation and analysis. By'topographic', we imply that the geometric structure of the data be optimally preserved in the transformation, and the embodiment of this constraint is that the inter-point distances in the feature space should correspond as closely as possible to those distances in the data space. The implementation of this principle by a neural network is very simple. A Radial Basis Function (RBF) neural network is utilised to predict the coordinates of the data point in the transformed feature space. The locations of the feature points are indirectly determined by adjusting the weights of the network. The transformation is determined by optimising the network parameters in order to minimise a suitable error measure that embodies the topographic principle. The specific details of this alternative approach are as follows.

To appear in Neural Computation.

Many real world patterns have a hierarchical underlying structure in which simple features have a higher order structure among themselves. Because these relationships are often statistical in nature, it is natural to view the process of discovering such structures as a statistical inference problem in which a hierarchical model is fit to data. Hierarchical statistical structure can be conveniently represented with Bayesian belief networks (Pearl, 1988; Lauritzen and Spiegelhalter, 1988; Neal, 1992). These 530 M. S. Lewicki and T. 1. Sejnowski models are powerful, because they can capture complex statistical relationships among the data variables, and also mathematically convenient, because they allow efficient computation of the joint probability for any given set of model parameters.