Statistical Learning Theory

The goal of statistical learning theory is to understand the learning mechanisms of modern machine learning algorithms and to use this knowledge for the design of new learning algorithms. In our group we have recently focussed on the following aspects:
 Kernelbased learning algorithms

Kernelbased learning algorithms such as support vector machines are a widelyspread class of offtheshelf learning algorithms, which often produce stateoftheart results for unstructured data sets. We have been investigating all aspects of these algorithms including their statistical analysis, their approximation capabilities, and efficient modifications for largescale data sets.
 Cluster analysis

Cluster analysis is a core task in data exploration. Among the several existing notions for clusters we have mainly focussed on the statistical analysis of densitybased clustering algorithms.
 Efficient learning algorithms

The computational requirements of many successful learning algorithms can become prohibitive for largescale data sets.We have been using our statistical insights to design new algorithms that require significantly less resources without sacrificing statistical guarantees.
 Loss functions and their risks

Most supervised learning algorithms optimize an objective function whose main ingredient is an empirical error term, where the error is measured with the help of a loss function. The choice of the loss function determines the learning goal and thus the outcome of the learning process. We have been interested in the relationship between loss functions and learning goals, asking, for example, which learning goals can be expressed by a loss function.
 Learning from non .i.i.d. data

The existing statistical analysis of machine learning algorithms assume in most cases independent, identically distributed (i.i.d.) data. In many cases, however, this assumption is violated. We have been focussing on learning from both dependent random data, and deterministic data.
 Reproducing kernel Hilbert spaces

Reproducing kernel Hilbert spaces do not only play a central role in kernelbased learning algorithms but are also an important notion in various other areas of mathematics. We have been investigating several aspects of these spaces and their interactions with probability measures.