In this paper we are concerned with reproducing kernel Hilbert spaces HK of functions from an input space into a Hilbert space Y, an environment appropriate for multi-task learning. The reproducing kernel K associated to HK has its values as operators on Y. Our primary goal here is to derive conditions which ensure that the kernel K is universal. This means that on every compact subset of the i...

Support vector regression (SVR) has been regarded as a state-of-the-art method for approximation and regression. The importance of kernel function, which is so-called admissible support vector kernel (SV kernel) in SVR, has motivated many studies on its composition. The Gaussian kernel (RBF) is regarded as a “best” choice of SV kernel used by non-expert in SVR, whereas there is no evidence, exc...

We demonstrate that a reproducing kernel Hilbert space of functions on a separable absolute Borel space or an analytic subset of a Polish space is separable if it possesses a Borel measurable feature map.

We demonstrate that a reproducing kernel Hilbert or Banach space of functions on a separable absolute Borel space or an analytic subset of a Polish space is separable if it possesses a Borel measurable feature map.

1 Lecture 1: Model Theory seca The basic idea behind model theory is to associate a Hilbert space construction with a function, and then use Hilbert space theory to illuminate the function theory. In one variable, one approach is to study the de Branges-Rovnyak space associated with a function φ in the ball of H(D). This is the Hilbert space of analytic functions on the disk D with reproducing ...

We propose a discretized Tikhonov regularization for a Cauchy problem for an elliptic equation by a reproducing kernel Hilbert space. We prove the convergence of discretized regularized solutions to an exact solution. Our numerical results demonstrate that our method can stably reconstruct solutions to the Cauchy problems even in severe cases of geometric configurations.

We study learning algorithms generated by regularization schemes in reproducing kernel Hilbert spaces associated with an -insensitive pinball loss. This loss function is motivated by the -insensitive loss for support vector regression and the pinball loss for quantile regression. Approximation analysis is conducted for these algorithms by means of a variance-expectation bound when a noise condi...