'Kernel' principal component analysis (PCA) is an elegant non(cid:173) linear generalisation of the popular linear data analysis method, where a kernel function implicitly defines a nonlinear transforma(cid:173) tion into a feature space wherein standard PCA is performed. Un(cid:173) fortunately, the technique is not'sparse', since the components thus obtained are expressed in terms of kernels associated with ev(cid:173) ery training vector. This paper shows that by approximating the covariance matrix in feature space by a reduced number of exam(cid:173) ple vectors, using a maximum-likelihood approach, we may obtain a highly sparse form of kernel PCA without loss of effectiveness.

The Vicinal Risk Minimization principle establishes a bridge between generative models and methods derived from the Structural Risk Mini(cid:173) mization Principle such as Support Vector Machines or Statistical Reg(cid:173) ularization. We explain how VRM provides a framework which inte(cid:173) grates a number of existing algorithms, such as Parzen windows, Support Vector Machines, Ridge Regression, Constrained Logistic Classifiers and Tangent-Prop. We then show how the approach implies new algorithm(cid:173) s for solving problems usually associated with generative models. New algorithms are described for dealing with pattern recognition problems with very different pattern distributions and dealing with unlabeled data. Preliminary empirical results are presented.

An active set strategy is applied to the dual of a simple reformula(cid:173) tion of the standard quadratic program of a linear support vector machine. This application generates a fast new dual algorithm that consists of solving a finite number of linear equations, with a typically large dimensionality equal to the number of points to be classified. However, by making novel use of the Sherman-Morrison(cid:173) Woodbury formula, a much smaller matrix of the order of the orig(cid:173) inal input space is inverted at each step. Thus, a problem with a 32-dimensional input space and 7 million points required inverting positive definite symmetric matrices of size 33 x 33 with a total run(cid:173) ning time of 96 minutes on a 400 MHz Pentium II. The algorithm requires no specialized quadratic or linear programming code, but merely a linear equation solver which is publicly available.

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 model(cid:173) can be used for quantifying the dependencies in graphical models.

In theory, the Winnow multiplicative update has certain advantages over the Perceptron additive update when there are many irrelevant attributes. Recently, there has been much effort on enhancing the Perceptron algo(cid:173) rithm by using regularization, leading to a class of linear classification methods called support vector machines. Similarly, it is also possible to apply the regularization idea to the Winnow algorithm, which gives meth(cid:173) ods we call regularized Winnows. We show that the resulting methods compare with the basic Winnows in a similar way that a support vector machine compares with the Perceptron. We investigate algorithmic is(cid:173) sues and learning properties of the derived methods.

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, typ(cid:173) ically binary-valued. The algorithm uses a learning procedure, which has access to n training examples, {(Xl, Y1), ..., (xn, Yn)}, drawn randomly from X x Yac(cid:173) cording to distribution D; it outputs a hypothesis I: X -t Y, whose error is the expected value of a loss function on I(x), g(x), where X is chosen according to D. Given f, cl 0 and access to random examples, a strong learning procedure outputs with probability 1 - cl a hypothesis with error at most f, with running time polyno(cid:173) mial in 1/ f, 1/ cl and the number of examples. A weak learning procedure satisfies the same conditions, but where f need only be better than random guessing. Schapire (1990) showed that any weak learning procedure, denoted WeakLeam, can be efficiently transformed ("boosted") into a strong learning procedure. The AdaBoost algorithm achieves this by calling WeakLeam multiple times, in a se(cid:173) quence of T stages, each time presenting it with a different distribution over a fixed training set and finally combining all of the hypotheses. The algorithm maintains a weight w: for each training example i at stage i, and a distribution D t is computed by normalizing these weights.

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.

A central issue in principal component analysis (PCA) is choosing the number of principal components to be retained. By interpreting PCA as density estimation, we show how to use Bayesian model selection to es(cid:173) timate the true dimensionality of the data. The resulting estimate is sim(cid:173) ple to compute yet guaranteed to pick the correct dimensionality, given enough data. The estimate involves an integral over the Steifel manifold of k-frames, which is difficult to compute exactly. But after choosing an appropriate parameterization and applying Laplace's method, an accu(cid:173) rate and practical estimator is obtained.

We present efficient algorithms for all-point-pairs problems, or'N(cid:173) body '-like problems, which are ubiquitous in statistical learning. We focus on six examples, including nearest-neighbor classification, kernel density estimation, outlier detection, and the two-point correlation. These include any problem which abstractly requires a comparison of each of the N points in a dataset with each other point and would naively be solved using N 2 distance computations. In practice N is often large enough to make this infeasible. We present a suite of new geometric t echniques which are applicable in principle to any'N-body' computation including large-scale mixtures of Gaussians, RBF neural networks, and HMM's.

A system has been developed to extract diagnostic information from jet engine carcass vibration data. Support Vector Machines applied to nov(cid:173) elty detection provide a measure of how unusual the shape of a vibra(cid:173) tion signature is, by learning a representation of normality. We describe a novel method for Support Vector Machines of including information from a second class for novelty detection and give results from the appli(cid:173) cation to Jet Engine vibration analysis.