Maximal margin classification for metric spaces

Abstract

In order to apply the maximum margin method in arbitrary metric spaces, we suggest to embed the metric space into a Banach or Hilbert space and to perform linear classification in this space. We propose several embeddings and recall that an isometric embedding in a Banach space is always possible while an isometric embedding in a Hilbert space is only possible for certain metric spaces. As a result, we obtain a general maximum margin classification algorithm for arbitrary metric spaces (whose solution is approximated by an algorithm of Graepel et al. (International Conference on Artificial Neural Networks 1999, pp. 304–309)). Interestingly enough, the embedding approach, when applied to a metric which can be embedded into a Hilbert space, yields the support vector machine (SVM) algorithm, which emphasizes the fact that its solution depends on the metric and not on the kernel. Furthermore, we give upper bounds of the capacity of the function classes corresponding to both embeddings in terms of Rademacher averages. Finally, we compare the capacities of these function classes directly.