A modification is described to the use of mean field approximations inthe E step of EM algorithms for analysing data from latent structure models, as described by Ghahramani (1995), among others. Themodification involves second-order Taylor approximations to expectations computed in the E step. The potential benefits of the method are illustrated using very simple latent profile models. 1 Introduction Ghahramani (1995) advocated the use of mean field methods as a means to avoid the heavy computation involved in the E step of the EM algorithm used for estimating parameters within a certain latent structure model, and Ghahramani & Jordan (1995) used the same ideas in a more complex situation. Dunmur & Titterington (1996a) identified Ghahramani's model as a so-called latent profile model, they observed that Zhang (1992,1993) had used mean field methods for a similar purpose, and they showed, in a simulation study based on very simple examples, that the mean field version of the EM algorithm often performed very respectably. By this it is meant that, when data were generated from the model under analysis, the estimators of the underlying parameters were efficient, judging by empirical results, especially in comparison with estimators obtained by employing the'correct' EM algorithm: the examples therefore had to be simple enough that the correct EM algorithm is numerically feasible, although any success reported for the mean field 432 A. P. Dunmur and D. M. Titterington version is, one hopes, an indication that the method will also be adequate in more complex situations in which the correct EM algorithm is not implementable because of computational complexity. In spite of the above positive remarks, there were circumstances in which there was a perceptible, if not dramatic, lack of efficiency in the simple (naive) mean field estimators, and the objective of this contribution is to propose and investigate ways of refining the method so as to improve performance without detracting from the appealing, and frequently essential, simplicity of the approach. The procedure used here is based on a second order correction to the naive mean field well known in statistical physics and sometimes called the cavity or TAP method (Mezard, Parisi & Virasoro, 1987). It has been applied recently in cluster analysis (Hofmann & Buhmann, 1996). In Section 2 we introduce the structure of our model, Section 3 explains the refined mean field approach, Section 4 provides numerical results, and Section 5 contains a statement of our conclusions.

By now it is widely accepted that learning a task from scratch, i.e., without any prior knowledge, is a daunting undertaking.

Constructing amulti-layered neural network with fixed architecture which implements orientation selectivity, we define output elements corresponding todifferent orientations, which allow us to make a selection decision. The algorithm takes into account the granularity of the lattice as well as the presence of noise and inefficiencies. The method is applied to a sample of data collected with the ZEUS detector at HERA in order to identify cosmic muons that leave a linear pattern of signals in the segmented calorimeter. A two dimensional representation of the relevant part of the detector is used. The algorithm performs very well. Given its architecture, this system becomes a good candidate for fast pattern recognition in parallel processing devices.

This paper describes a new technique for object recognition based on learning appearance models. The image is decomposed into local regions which are described by a new texture representation called "Generalized Second Moments" thatare derived from the output of multiscale, multiorientation filter banks. Class-characteristic local texture features and their global composition is learned by a hierarchical mixture of experts architecture (Jordan & Jacobs). The technique is applied to a vehicle database consisting of 5 general car categories (Sedan, Van with backdoors, Van without backdoors, old Sedan, and Volkswagen Bug). This is a difficult problem with considerable in-class variation. The new technique has a 6.5% misclassification rate, compared to eigen-images which give 17.4% misclassification rate, and nearest neighbors which give 15 .7%

This paper compares three penalty terms with respect to the efficiency ofsupervised learning, by using first-and second-order learning algorithms. Our experiments showed that for a reasonably adequate penaltyfactor, the combination of the squared penalty term and the second-order learning algorithm drastically improves the convergence performance more than 20 times over the other combinations, atthe same time bringing about a better generalization performance.

We cast the problem as one of maximum likelihood density estimation, andin that framework introduce an algorithm that searches for independent components using both temporal and spatial cues. We call the resulting algorithm "Contextual ICA," after the (Bell and Sejnowski 1995) Infomax algorithm, which we show to be a special case of cICA. Because cICA can make use of the temporal structure of its input, it is able separate in a number of situations where standard methods cannot, including sources with low kurtosis, coloredGaussian sources, and sources which have Gaussian histograms. 1 The Blind Source Separation Problem Consider a set of n indepent sources

In the present paper, we propose a method to unify information maximization and minimization in hidden units. The information maximization and minimization are performed on two different levels: collectiveand individual level. Thus, two kinds of information: collective and individual information are defined. By maximizing collective information and by minimizing individual information, simple networks can be generated in terms of the number of connections andthe number of hidden units. Obtained networks are expected to give better generalization and improved interpretation of internal representations.

A classifier is called consistent with respect to a given set of classlabeled pointsif it correctly classifies the set. We consider classifiers defined by unions of local separators and propose algorithms for consistent classifier reduction. The expected complexities of the proposed algorithms are derived along with the expected classifier sizes. In particular, the proposed approach yields a consistent reduction ofthe nearest neighbor classifier, which performs "firm" classification, assigning each new object to a class, regardless of the data structure. The proposed reduction method suggests a notion of "soft" classification, allowing for indecision with respect to objects which are insufficiently or ambiguously supported by the data. The performances of the proposed classifiers in predicting stockbehavior are compared to that achieved by the nearest neighbor method. 1 Introduction Certain classification problems, such as recognizing the digits of a hand written zipcode, requirethe assignment of each object to a class. Others, involving relatively small amounts of data and high risk, call for indecision until more data become available. Examples in such areas as medical diagnosis, stock trading and radar detection are well known. The training data for the classifier in both cases will correspond to firmly labeled members of the competing classes.

We consider the microscopic equations for learning problems in neural networks. The aligning fields of an example are obtained from the cavity fields, which are the fields if that example were absent in the learning process. In a rough energy landscape, we assume that the density of the local minima obey an exponential distribution, yielding macroscopic properties agreeing with the first step replica symmetry breaking solution. Iterating the microscopic equations provide a learning algorithm, which results in a higher stability than conventional algorithms. 1 INTRODUCTION Most neural networks learn iteratively by gradient descent. As a result, closed expressions forthe final network state after learning are rarely known. This precludes further analysis of their properties, and insights into the design of learning algorithms.

This paper investigates the stationary points of a Hebb learning rule with a sigmoid nonlinearity in it. We show mathematically that when the input has a low information content, as measured by the input's variance, this learning rule suppresses learning, that is, forces the weight vector to converge to the zero vector. When the information content exceeds a certain value, the rule will automatically begin to learn a feature in the input. Our analysis suggests that under certain conditions it is the first principal component that is learned. The weight vector length remains bounded, provided the variance of the input is finite. Simulations confirm the theoretical results derived.