Positive definite multi-kernels for scattered data interpolations

Abstract

In this article, we use the knowledge of positive definite tensors to develop a concept of positive definite multi-kernels to construct the kernel-based interpolants of scattered data. By the techniques of reproducing kernel Banach spaces, the optimal recoveries and error analysis of the kernel-based interpolants are shown for a special class of strictly positive definite multi-kernels.

Introduction

In many areas of practical applications, we often face a problem of reconstructing an unknown function f from the scattered data. The scattered data consist of high-dimension data points ${\left\{{\mathit{x}}_i\right\}}_{i=1}^n$ and data values ${\left\{y_i\right\}}_{i=1}^n$ such that $y_i=f({\mathit{x}}_i)$ for all $i=1,2,\dots ,n$. The reconstruction is to find an estimate function s to approximate f. Generally, s is sought to interpolate the scattered data, that is, $s({\mathit{x}}_i)=y_i$ for all $i=1,2,\dots ,n$. It is well-known that the kernel-based approximation method is a fundamental approach of scattered data interpolations. The classical kernel-based approximation method is mainly dependent of the positive definite kernels. Here, we develop a concept of positive definite multi-kernels in Definition 3.1 which is a generalization of positive definite kernels. The positive definite multi-kernels can be also used to reconstruct f from the scattered data. For examples, we compare the interpolations by a positive definite kernel $K_2:{\otimes }_{k=1}^2{\mathbb{R}}^d\to \mathbb{R}$ and a positive definite multi-kernel $K_4:{\otimes }_{k=1}^4{\mathbb{R}}^d\to \mathbb{R}$. The classical interpolant $s_2$ is composed of a kernel basis

$$K_2(\cdot ,{\mathit{x}}_i),\text{for }i=1,2,\dots ,n,$$

that is,

$$s_2(\mathit{x}):=\sum _{i=1}^n{c}_i{K}_2(\mathit{x},{\mathit{x}}_i),\text{for }\mathit{x}\in {\mathbb{R}}^d,$$

where the coefficients $c_1,c_2,\dots ,c_n$ are solved by a linear system

$$\sum _{i_2=1}^n{K}_2({\mathit{x}}_{i_1},{\mathit{x}}_{i_2})c_{i_2}=y_{i_1},\text{for }{i}_1=1,2,\dots ,n.$$

The new interpolant $s_4$ is composed of another kernel basis

$$K_4(\cdot ,{\mathit{x}}_{i_1},{\mathit{x}}_{i_2},{\mathit{x}}_{i_3}),\text{for }{i}_1,i_2,i_3=1,2,\dots ,n,$$

that is,

$$s_4(\mathit{x}):=\sum _{i_1,i_2,i_3=1}^{n,n,n}{c}_{i_1}{c}_{i_2}{c}_{i_3}{K}_4(\mathit{x},{\mathit{x}}_{i_1},{\mathit{x}}_{i_2},{\mathit{x}}_{i_3}),\text{for }\mathit{x}\in {\mathbb{R}}^d,$$

where the coefficients $c_1,c_2,\dots ,c_n$ are solved by a multi-linear system

$$\sum _{i_2,i_3,i_4=1}^{n,n,n}{K}_4({\mathit{x}}_{i_1},{\mathit{x}}_{i_2},{\mathit{x}}_{i_3},{\mathit{x}}_{i_4})c_{i_2}{c}_{i_3}{c}_{i_4}=y_{i_1},\text{for }{i}_1=1,2,\dots ,n.$$

In this article, we mainly discuss how to use a special class of strictly positive definite multi-kernel ${\mathrm{\Phi }}_m$ in Equation (3.3) to construct the kernel-based interpolant $s_m$ in Theorem 4.1 which shows that the related multi-linear system exists the unique solution. By the theorems of reproducing kernel Banach spaces in [5], we can obtain the advanced properties of $s_m$ including optimal recoveries in Theorem 4.3 and error analysis in Theorem 4.5, Theorem 4.6.