Kernel Methods in Machine Learning 1

We review machine learning methods employing positive definite kernels. These methods formulate learning and estimation problems in a reproducing kernel Hilbert space (RKHS) of functions defined on the data domain, expanded in terms of a kernel. Working in linear spaces of function has the benefit of facilitating the construction and analysis of learning algorithms while at the same time allowing large classes of functions. The latter include nonlinear functions as well as functions defined on nonvectorial data. We cover a wide range of methods, ranging from binary classifiers to sophisticated methods for estimation with structured data. 1. Introduction. Over the last ten years estimation and learning methods utilizing positive definite kernels have become rather popular, particularly in machine learning. Since these methods have a stronger mathematical slant than earlier machine learning methods (e.g., neural networks), there is also significant interest in the statistics and mathematics community for these methods. The present review aims to summarize the state of the art on a conceptual level. In doing so, we build on various sources, including Burges but we also add a fair amount of more recent material which helps unifying the exposition. We have not had space to include proofs; they can be found either in the long version of the present paper (see Hofmann et al. [69]), in the references given or in the above books. The main idea of all the described methods can be summarized in one paragraph. Traditionally, theory and algorithms of machine learning and