Learning with Non Positive Kernels
Cheng Soon Ong - Australian National University
Xavier Mary - ENSAE-CREST-LS
Stephane Canu - INSA de Rouen
Alexander Smola - Australian National University and NICTA
In this paper we show that many kernel methods can be adapted to deal with indefinite kernels, that is, kernels which are not positive semidefinite. They do not satisfy Mercer's condition and they induce associated functional spaces called Reproducing Kernel Krein Spaces (RKKS), a generalization of Reproducing Kernel Hilbert Spaces (RKHS). Machine learning in RKKS shares many ``nice'' properties of learning in RKHS, such as orthogonality and projection. However, since the kernels are indefinite, we can no longer minimize the loss, instead we stabilize it. We show a general representer theorem for constrained stabilization and prove generalization bounds by computing the Rademacher averages of the kernel class. We list several examples of indefinite kernels and investigate regularization methods to solve spline interpolation. Some preliminary experiments with indefinite kernels for spline smoothing are reported for truncated spectral factorization, Landweber-Fridman iterations, and MR-II.