Abstract
Kernel based regularized interpolation is one of the most important methods for approximating functions. The theory behind the kernel based regularized interpolation is the well-known Representer Theorem, which shows the form of approximation function in the reproducing kernel Hilbert spaces. Because of the advantages of the kernel based regularized interpolation, it is widely used in many mathematical and engineering applications, for example, dimension reduction and dimension estimation. However, the performance of the approximation is not fully understood from the theoretical perspective. In other word, the error analysis for the kernel based regularized interpolation is lacking. In this paper, some error bounds in terms of the reproducing kernel Hilbert space norm and Sobolev space norm are given to understand the behavior of the approximation function.
Highlights
Approximating functions in high dimensional spaces is one of the central problems in both mathematics and engineering
The theory behind the kernel based regularized interpolation is the well-known Representer Theorem, which shows the form of approximation function in the reproducing kernel Hilbert spaces
Some error bounds in terms of the reproducing kernel Hilbert space norm and Sobolev space norm are given to understand the behavior of the approximation function
Summary
Approximating functions in high dimensional spaces is one of the central problems in both mathematics and engineering. When only a few samples are given, Kriging will yield non-stationary behavior [10] Another alternative for approximating multivariate functions is the inverse distance weighting (IDW) method, which was originally proposed in [9]. The kernel based regularized interpolation works for the case that the observations have some noise (see Section 2 for analysis). This is one of the biggest advantages that other interpolation methods do not have. We would like to provide some error analysis for the kernel based regularized interpolation from the function space point of view. We focus on deriving Hilbert space type and Sobolev space type error estimates for the kernel based regularized interpolation
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