We present a generalization of the adversarial linear bandits framework, where the underlying losses are kernel functions (with an associated reproducing kernel Hilbert space) rather than linear functions. We study a version of the exponential weights algorithm and bound its regret in this setting. Under conditions on the eigen-decay of the kernel we provide a sharp characterization of the regr...

In a previous paper, we developed a general framework for establishing tractability and strong tractability for quasilinear multivariate problems in the worst case setting. One important example of such a problem is the solution of the heat equation ut = 1u − qu in I d × (0, T ), where I is the unit interval and T is a maximum time value. This problem is to be solved subject to homogeneous Diri...

A lot of problems the physical world can be modeled by non-linear ODE with their initial and boundary conditions. Especially higher order differential equations play a vital role in this process. The method for solution its effectiveness are as important modelling. In paper, on basis reproducing kernel theory, functions have been obtained solving some equations. Additionally, each problem homog...

Conditionally positive definite kernels provide a powerful tool for scattered data approximation. Many nice properties of such methods follow from an underlying reproducing kernel structure. While the connection between positive definite kernels and reproducing kernel Hilbert spaces is well understood, the analog relation between conditionally positive definite kernels and reproducing kernel Po...

We consider the global minimization of a polynomial on compact set $$\mathbf {B}$$ . show that each step Moment-SOS hierarchy has nice and simple interpretation complements usual one. Namely, it computes coefficients in an orthonormal basis $$L^2(\mathbf {B},\mu )$$ where $$\mu $$ is arbitrary reference measure whose support exactly The resulting certain density (with respect to ) some signed W...

The reproducing kernel function of a weighted Bergman space over domains in C is known explicitly in only a small number of instances. Here, we introduce a process of orthogonal norm expansion along a subvariety of codimension 1, which also leads to a series expansion of the reproducing kernel in terms of reproducing kernels defined on the subvariety. The problem of finding the reproducing kern...

Abstract In this chapter we review some basic ideas for nonlinear system identification. This is a complex area with vast and rich literature. One reason the richness that very many parameterizations of unknown have been suggested, each various proposed estimation methods. We will first describe details nonparametric techniques based on Reproducing Kernel Hilbert Space theory Gaussian regressio...