Geometrical aspects of statistical learning theory

Geometry plays an important role in modern statistical learning theory, and many different aspects of geometry can be found in this fast developing field. This thesis addresses some of these aspects. A large part of this work will be concerned with so called manifold methods, which have recently attracted a lot of interest. The key point is that for a lot of real-world data sets it is natural to assume that the data lies on a low-dimensional submanifold of a potentially high-dimensional Euclidean space. We develop a rigorous and quite general framework for the estimation and approximation of some geometric structures and other quantities of this submanifold, using certain corresponding structures on neighborhood graphs built from random samples of that submanifold. Another part of this thesis deals with the generalization of the maximal margin principle to arbitrary metric spaces. This generalization follows quite naturally by changing the viewpoint on the well-known support vector machines (SVM). It can be shown that the SVM can be seen as an algorithm which applies the maximum margin principle to a subclass of metric spaces. The motivation to consider the generalization to arbitrary metric spaces arose by the observation that in practice the condition for the applicability of the SVM is rather difficult to check for a given metric. Nevertheless one would like to apply the successful maximum margin principle even in cases where the SVM cannot be applied. The last part deals with the specific construction of so called Hilbertian metrics and positive definite kernels on probability measures. We consider several ways of building such metrics and kernels. The emphasis lies on the incorporation of different desired properties of the metric and kernel. Such metrics and kernels have a wide applicability in so called kernel methods since probability measures occur as inputs in various situations.