Clifford Analysis with Indefinite Signature

Signature Change and Clifford Algebras

Given the real Clifford algebra of a quadratic space with a given signature, we define a new product in this structure such that it simulates the Clifford product of a quadratic space with another signature different from the original one. Among the possible applications of this new product, we use it in order to write the minkowskian Dirac equation over the euclidean spacetime and to define a ...

The Clifford-Klein Space Forms of Indefinite Metric

3 Introductory Clifford Analysis

We want in this chapter to regard Clifford algebras as natural generalizations of the complex number system. First let us note that if z is a complex number then zz = ‖z‖. Now for a quaternion q we also have qq = ‖q‖. In this way quaternions may be regarded as a generalization of the complex number system. It seems natural to ask if one can extend basic results of one complex variable analysis ...

Analysis of SVM with Indefinite Kernels

The recent introduction of indefinite SVM by Luss and d’Aspremont [15] has effectively demonstrated SVM classification with a non-positive semi-definite kernel (indefinite kernel). This paper studies the properties of the objective function introduced there. In particular, we show that the objective function is continuously differentiable and its gradient can be explicitly computed. Indeed, we ...

Indefinite Kernel Discriminant Analysis

Kernel methods for data analysis are frequently considered to be restricted to positive definite kernels. In practice, however, indefinite kernels arise e.g. from problem-specific kernel construction or optimized similarity measures. We, therefore, present formal extensions of some kernel discriminant analysis methods which can be used with indefinite kernels. In particular these are the multi-...