Learnability in Hilbert Spaces with Reproducing Kernels

Radial kernels and their reproducing kernel Hilbert spaces

We describe how to use Schoenberg’s theorem for a radial kernel combined with existing bounds on the approximation error functions for Gaussian kernels to obtain a bound on the approximation error function for the radial kernel. The result is applied to the exponential kernel and Student’s kernel. To establish these results we develop a general theory regarding mixtures of kernels. We analyze t...

Reproducing Kernels for Hilbert Spaces of Real Harmonic Functions

This paper studies a family of Hilbert spaces of real harmonic functions on bounded regions in Rn and will show that, for a range of values of s, they are reproducing kernel Hilbert spaces. The spaces are characterized by their boundary traces and the inner products are defined via their expansions in the harmonic Steklov eigenfunctions of the region. The reproducing kernels will then be descri...

Real reproducing kernel Hilbert spaces

P (α) = C(α, F (x, y)) = αF (x, x) + 2αF (x, y) + F (x, y)F (y, y), which is ≥ 0. In the case F (x, x) = 0, the fact that P ≥ 0 implies that F (x, y) = 0. In the case F (x, y) 6= 0, P (α) is a quadratic polynomial and because P ≥ 0 it follows that the discriminant of P is ≤ 0: 4F (x, y) − 4 · F (x, x) · F (x, y)F (y, y) ≤ 0. That is, F (x, y) ≤ F (x, y)F (x, x)F (y, y), and this implies that F ...

Online learning in Reproducing Kernel Hilbert Spaces

Bayesian Learning in Reproducing Kernel Hilbert Spaces

Support Vector Machines nd the hypothesis that corresponds to the centre of the largest hypersphere that can be placed inside version space, i.e. the space of all consistent hypotheses given a training set. The boundaries of version space touched by this hypersphere de ne the support vectors. An even more promising approach is to construct the hypothesis using the whole of version space. This i...