Adaptive boosting methods are simple modular algorithms that operate as follows. Let 9: X -t Y be the function to be learned, where the label set Y is finite, typically binary-valued. The algorithm uses a learning procedure, which has access to n training examples, {(Xl, Y1),..., (xn, Yn)}, drawn randomly from X x Yaccording to distribution D; it outputs a hypothesis I:

We introduce the mixture of Gaussian processes (MGP) model which is useful for applications in which the optimal bandwidth of a map is input dependent. The MGP is derived from the mixture of experts model and can also be used for modeling general conditional probability densities. We discuss how Gaussian processes -in particular in form of Gaussian process classification, the support vector machine and the MGP modelcan be used for quantifying the dependencies in graphical models. 1 Introduction Gaussian processes are typically used for regression where it is assumed that the underlying function is generated by one infinite-dimensional Gaussian distribution (i.e.

Two well known classes of unsupervised procedures that can be cast in this manner are generative and recoding models. In a generative unsupervised framework, the environment generates training exampleswhich we will refer to as observations-by sampling from one distribution; the other distribution is embodied in the model. Examples of generative frameworks are mixtures of Gaussians (MoG) [2], factor analysis [4], and Boltzmann machines [8]. In the recoding unsupervised framework, the model transforms points from an obser- vation space to an output space, and the output distribution is compared either to a reference distribution or to a distribution derived from the output distribution. An example is independent component analysis (leA) [11], a method that discovers a representation of vector-valued observations in which the statistical dependence among the vector elements in the output space is minimized.

We introduce a method of feature selection for Support Vector Machines. The method is based upon finding those features which minimize bounds on the leave-one-out error. This search can be efficiently performed via gradient descent. The resulting algorithms are shown to be superior to some standard feature selection algorithms on both toy data and real-life problems of face recognition, pedestrian detection and analyzing DNA micro array data.

Classification is achieved by a linear or nonlinear separating surface in the input space of the dataset. In this work we propose a very fast simple algorithm, based on an active set strategy for solving quadratic programs with bounds [18]. The algorithm is capable of accurately solving problems with millions of points and requires nothing more complicated than a commonly available linear equation solver [17, 1, 6] for a typically small (100) dimensional input space of the problem. Key to our approach are the following two changes to the standard linear SVM: 1. Maximize the margin (distance) between the parallel separating planes with respect to both orientation (w) as well as location relative to the origin b).

Vapnik's result that the expectation of the generalisation error ofthe optimal hyperplane is bounded by the expectation of the ratio of the number of support vectors to the number of training examples is extended to a broad class of kernel machines. The class includes Support Vector Machines for soft margin classification and regression, and Regularization Networks with a variety of kernels and cost functions. We show that key inequalities in Vapnik's result become equalities once "the classification error" is replaced by "the margin error", with the latter defined as an instance with positive cost. In particular we show that expectations of the true margin error and the empirical margin error are equal, and that the sparse solutions for kernel machines are possible only if the cost function is "partially" insensitive. 1 Introduction Minimization of regularized risk is a backbone of several recent advances in machine learning, including Support Vector Machines (SVM) [13], Regularization Networks (RN) [5] or Gaussian Processes [15]. Such a machine is typically implemented as a weighted sum of a kernel function evaluated for pairs composed of a data vector in question and a number of selected training vectors, so called support vectors.

We present a bound on the generalisation error of linear classifiers in terms of a refined margin quantity on the training set. The result is obtained in a PAC-Bayesian framework and is based on geometrical arguments in the space of linear classifiers. The new bound constitutes an exponential improvement of the so far tightest margin bound by Shawe-Taylor et al. [8] and scales logarithmically in the inverse margin. Even in the case of less training examples than input dimensions sufficiently large margins lead to nontrivial bound values and - for maximum margins - to a vanishing complexity term. Furthermore, the classical margin is too coarse a measure for the essential quantity that controls the generalisation error: the volume ratio between the whole hypothesis space and the subset of consistent hypotheses.

'Kernel' principal component analysis (PCA) is an elegant nonlinear generalisation of the popular linear data analysis method, where a kernel function implicitly defines a nonlinear transformation into a feature space wherein standard PCA is performed. Unfortunately, the technique is not'sparse', since the components thus obtained are expressed in terms of kernels associated with every training vector. This paper shows that by approximating the covariance matrix in feature space by a reduced number of example vectors, using a maximum-likelihood approach, we may obtain a highly sparse form of kernel PCA without loss of effectiveness. 1 Introduction Principal component analysis (PCA) is a well-established technique for dimensionality reduction, and examples of its many applications include data compression, image processing, visualisation, exploratory data analysis, pattern recognition and time series prediction.

These include face detection [14], face pose discrimination [12] and face recognition [16]. Although facial sex classification has attracted much attention in the psychological literature [1, 4, 8, 15], relatively few computatinal learning methods have been proposed. We will briefly review and summarize the prior art in facial sex classification.

Learning of a smooth but nonparametric probability density can be regularized using methods of Quantum Field Theory. We implement a field theoretic prior numerically, test its efficacy, and show that the free parameter of the theory (,smoothness scale') can be determined self consistently by the data; this forms an infinite dimensional generalization of the MDL principle. Finally, we study the implications of one's choice of the prior and the parameterization and conclude that the smoothness scale determination makes density estimation very weakly sensitive to the choice of the prior, and that even wrong choices can be advantageous for small data sets. One of the central problems in learning is to balance'goodness of fit' criteria against the complexity of models. An important development in the Bayesian approach was thus the realization that there does not need to be any extra penalty for model complexity: if we compute the total probability that data are generated by a model, there is a factor from the volume in parameter space-the'Occam factor' -that discriminates against models with more parameters [1, 2].