Subspace Learning in Krein Spaces: Complete Kernel Fisher Discriminant Analysis with Indefinite Kernels

Positive definite kernels, such as Gaussian Radial Basis Functions (GRBF), have been widely used in computer vision for designing feature extraction and classification algorithms. In many cases nonpositive definite (npd) kernels and non metric similarity/dissimilarity measures naturally arise (e.g., Hausdorff distance, Kullback Leibler Divergences and Compact Support (CS) Kernels). Hence, there is a practical and theoretical need to properly handle npd kernels within feature extraction and classification frameworks. Recently, classifiers such as Support Vector Machines (SVMs) with npd kernels, Indefinite Kernel Fisher Discriminant Analysis (IKFDA) and Indefinite Kernel Quadratic Analysis (IKQA) were proposed. In this paper we propose feature extraction methods using indefinite kernels. In particular, first we propose an Indefinite Kernel Principal Component Analysis (IKPCA). Then, we properly define optimization problems that find discriminant projections with indefinite kernels and propose a Complete Indefinite Kernel Fisher Discriminant Analysis (CIKFDA) that solves the proposed problems. We show the power of the proposed frameworks in a fully automatic face recognition scenario.