The idea of a large minimum margin [17] explains the good generalization performance of AdaBoost in the low noise regime. However, AdaBoost performs worse on noisy tasks [10, 11], such as the iris and the breast cancer benchmark data sets [1]. On the latter tasks, a large margin on all training points cannot be achieved without adverse effects on the generalization error. This experimental observation was supported by the study of [13] where the generalization error of ensemble methods was bounded by the sum of the fraction of training points which have a margin smaller than some value p, say, plus a complexity term depending on the base hypotheses and p. While this bound can only capture part of what is going on in practice, it nevertheless already conveys the message that in some cases it pays to allow for some points which have a small margin, or are misclassified, if this leads to a larger overall margin on the remaining points. To cope with this problem, it was mandatory to construct regularized variants of AdaBoost, which traded off the number of margin errors and the size of the margin 562 G. Riitsch, B. Sch6lkopf, A. J. Smola, K.-R.

Suppose there exists a function y* fo(x) from which we observe the measurements corrupted with noise ((Xl, YI),".

The nonnegative Boltzmann machine (NNBM) is a recurrent neural network model that can describe multimodal nonnegative data. Application of maximum likelihood estimation to this model gives a learning rule that is analogous to the binary Boltzmann machine. We examine the utility of the mean field approximation for the NNBM, and describe how Monte Carlo sampling techniques can be used to learn its parameters. Reflective slice sampling is particularly well-suited for this distribution, and can efficiently be implemented to sample the distribution. We illustrate learning of the NNBM on a transiationally invariant distribution, as well as on a generative model for images of human faces. Introduction The multivariate Gaussian is the most elementary distribution used to model generic data. It represents the maximum entropy distribution under the constraint that the mean and covariance matrix of the distribution match that of the data. For the case of binary data, the maximum entropy distribution that matches the first and second order statistics of the data is given by the Boltzmann machine [1].

This question was recently shown in [9] to be a special case of a much more fundamental problem: What are the features of the variable X that are relevant for the prediction of another, relevance, variable Y?

We show that the recently proposed variant of the Support Vector machine (SVM) algorithm, known as v-SVM, can be interpreted as a maximal separation between subsets of the convex hulls of the data, which we call soft convex hulls. The soft convex hulls are controlled by choice of the parameter v. If the intersection of the convex hulls is empty, the hyperplane is positioned halfway between them such that the distance between convex hulls, measured along the normal, is maximized; and if it is not, the hyperplane's normal is similarly determined by the soft convex hulls, but its position (perpendicular distance from the origin) is adjusted to minimize the error sum. The proposed geometric interpretation of v-SVM also leads to necessary and sufficient conditions for the existence of a choice of v for which the v-SVM solution is nontrivial. 1 Introduction Recently, SchOlkopf et al. [I) introduced a new class of SVM algorithms, called v-SVM, for both regression estimation and pattern recognition. The basic idea is to remove the user-chosen error penalty factor C that appears in SVM algorithms by introducing a new variable p which, in the pattern recognition case, adds another degree of freedom to the margin.

We provide preliminary evidence that eXlstmg algorithms for inferring small-scale gene regulation networks from gene expression data can be adapted to large-scale gene expression data coming from hybridization microarrays. The essential steps are (1) clustering many genes by their expression time-course data into a minimal set of clusters of co-expressed genes, (2) theoretically modeling the various conditions under which the time-courses are measured using a continious-time analog recurrent neural network for the cluster mean time-courses, (3) fitting such a regulatory model to the cluster mean time courses by simulated annealing with weight decay, and (4) analysing several such fits for commonalities in the circuit parameter sets including the connection matrices. This procedure can be used to assess the adequacy of existing and future gene expression time-course data sets for determ ining transcriptional regulatory relationships such as coregulation.

Fishers linear discriminant analysis (LDA) is a classical multivariate technique both for dimension reduction and classification. The data vectors are transformed into a low dimensional subspace such that the class centroids are spread out as much as possible. In this subspace LDA works as a simple prototype classifier with linear decision boundaries. However, in many applications the linear boundaries do not adequately separate the classes. We present a nonlinear generalization of discriminant analysis that uses the kernel trick of representing dot products by kernel functions.

However, most models have simply ignored these data; the few that have attempted to address them have failed by at least an order of magnitude. We discuss key data on the speed of acquisition, and show how to account for them using a statistically sound model of learning, in which differential reliabilities of stimuli playa crucial role. 1 Introduction Conditioning experiments probe the ways that animals make predictions about rewards and punishments and how those predictions are used to their advantage. Substantial quantitative data are available as to how pigeons and rats acquire conditioned responses during autoshaping, which is one of the simplest paradigms of classical conditioning.

We consider the problem of reliably choosing a near-best strategy from a restricted class of strategies TI in a partially observable Markov decision process (POMDP). We assume we are given the ability to simulate the POMDP, and study what might be called the sample complexity - that is, the amount of data one must generate in the POMDP in order to choose a good strategy. We prove upper bounds on the sample complexity showing that, even for infinitely large and arbitrarily complex POMDPs, the amount of data needed can be finite, and depends only linearly on the complexity of the restricted strategy class TI, and exponentially on the horizon time. This latter dependence can be eased in a variety of ways, including the application of gradient and local search algorithms.

A new method for multivariate density estimation is developed based on the Support Vector Method (SVM) solution of inverse ill-posed problems. The solution has the form of a mixture of densities. This method with Gaussian kernels compared favorably to both Parzen's method and the Gaussian Mixture Model method. For synthetic data we achieve more accurate estimates for densities of 2, 6, 12, and 40 dimensions. 1 Introduction The problem of multivariate density estimation is important for many applications, in particular, for speech recognition [1] [7]. When the unknown density belongs to a parametric set satisfying certain conditions one can estimate it using the maximum likelihood (ML) method. Often these conditions are too restrictive. Therefore, nonparametric methods were proposed. The most popular of these, Parzen's method [5], uses the following estimate given data