SVM in Krĕın spaces

Support vector machines (SVM) and kernel methods have been highly successful in many application areas. However, the requirement that the kernel is symmetric positive semidefinite, Mercer’s condition, is not always verified in practice. When it is not, the kernel is called indefinite. Various heuristics and specialized methods have been proposed to address indefinite kernels, from simple tricks such as removing negative eigenvalues, to advanced methods that de-noise the kernel by considering the negative part of the kernel as noise. Most approaches aim at correcting an indefinite kernel in order to provide a positive one. We propose a new SVM approach that deals directly with indefinite kernels. In contrast to previous approaches, we embrace the underlying idea that the negative part of an indefinite kernel may contain valuable information. To define such a method, the SVM formulation has to be adapted to a non usual form: the stabilization. The hypothesis space, usually a Hilbert space, becomes a Krĕın space. This work explores this new formulation, and proposes two practical algorithms (ESVM and KSVM) that outperform the approaches that modify the kernel. Moreover, the solution depends on the original kernel and thus can be used on any new point without loss of accuracy. 1 ha l-0 08 69 65 8, v er si on 1 3 O ct 2 01 3