Stochastic processes with sample paths in reproducing kernel Hilbert spaces

Stochastic Processes with Sample Paths in Reproducing Kernel Hilbert Spaces

A theorem of M. F. Driscoll says that, under certain restrictions, the probability that a given Gaussian process has its sample paths almost surely in a given reproducing kernel Hilbert space (RKHS) is either 0 or 1. Driscoll also found a necessary and sufficient condition for that probability to be 1. Doing away with Driscoll’s restrictions, R. Fortet generalized his condition and named it nuc...

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

Distribution Embeddings in Reproducing Kernel Hilbert Spaces

The “kernel trick” is well established as a means of constructing nonlinear algorithms from linear ones, by transferring the linear algorithms to a high dimensional feature space: specifically, a reproducing kernel Hilbert space (RKHS). Recently, it has become clear that a potentially more far reaching use of kernels is as a linear way of dealing with higher order statistics, by embedding proba...

Quantile Regression in Reproducing Kernel Hilbert Spaces

In this paper we consider quantile regression in reproducing kernel Hilbert spaces, which we refer to as kernel quantile regression (KQR). We make three contributions: (1) we propose an efficient algorithm that computes the entire solution path of the KQR, with essentially the same computational cost as fitting one KQR model; (2) we derive a simple formula for the effective dimension of the KQR...