Backward shift invariant subspaces in reproducing kernel Hilbert spaces

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 ...

Invariant Subspaces for the Backward Shift on Hilbert Spaces of Analytic Functions with Regular Norm

We investigate the structure of invariant subspaces of backward shift operator Lf = (f − f(0))/ζ on a large class of abstract Hilbert spaces of analytic functions on the unit disc where the forward shift operator Mζf = ζf acts as a contraction. Our main results show that under certain regularity conditions on the norm of such a space, the functions in a nontrivial invariant subspace of L have m...

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...