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Product kernels are efficient and flexible tools for high-dimensional scattered data interpolation

Kristof Albrecht, Juliane Entzian, Armin Iske

TL;DR

This work analyzes the general problem of multivariate interpolation by product kernels and investigates the tensor product structure, in particular for grid-like samples, and develops an efficient computation scheme for the well-known Newton basis.

Abstract

This work concerns the construction and characterization of product kernels for multivariate approximation from a finite set of discrete samples. To this end, we consider composing different component kernels, each acting on a low-dimensional Euclidean space. Due to Aronszajn (1950), the product of positive semi-definite kernel functions is again positive semi-definite, where, moreover, the corresponding native space is a particular instance of a tensor product, referred to as Hilbert tensor product. We first analyze the general problem of multivariate interpolation by product kernels. Then, we further investigate the tensor product structure, in particular for grid-like samples. We use this case to show that the product of positive definite kernel functions is again positive definite. Moreover, we develop an efficient computation scheme for the well-known Newton basis. Supporting numerical examples show the good performance of product kernels, especially for their flexibility.

Product kernels are efficient and flexible tools for high-dimensional scattered data interpolation

TL;DR

This work analyzes the general problem of multivariate interpolation by product kernels and investigates the tensor product structure, in particular for grid-like samples, and develops an efficient computation scheme for the well-known Newton basis.

Abstract

This work concerns the construction and characterization of product kernels for multivariate approximation from a finite set of discrete samples. To this end, we consider composing different component kernels, each acting on a low-dimensional Euclidean space. Due to Aronszajn (1950), the product of positive semi-definite kernel functions is again positive semi-definite, where, moreover, the corresponding native space is a particular instance of a tensor product, referred to as Hilbert tensor product. We first analyze the general problem of multivariate interpolation by product kernels. Then, we further investigate the tensor product structure, in particular for grid-like samples. We use this case to show that the product of positive definite kernel functions is again positive definite. Moreover, we develop an efficient computation scheme for the well-known Newton basis. Supporting numerical examples show the good performance of product kernels, especially for their flexibility.
Paper Structure (12 sections, 15 theorems, 84 equations, 8 figures)

This paper contains 12 sections, 15 theorems, 84 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Product kernels
  3. Native spaces of product kernels
  4. Interpolation with product kernels
  5. Interpolation on grid-like data
  6. Stability
  7. Tensor Newton basis
  8. Componentwise point selection
  9. Numerical results
  10. Numerical stability versus approximation behaviour
  11. Computional efficiency
  12. Conclusion and Future Research

Key Result

Theorem 2.2

Let $K_i$ be positive semi-definite kernels on $\mathbb{R}^{d_i}$, $i=1,\dots,M$. Then, the product kernel $K = \prod_{i=1}^{M} K_i$ is positive semi-definite on $\mathbb{R}^d$, where $d = \sum_{i=1}^{M} d_i$. ∎

Figures (8)

  • Figure 1: Bivariate product kernel: Visualization of the univariate component kernels (left, middle) and top view on the product kernel (right).
  • Figure 2: Scattered points $X$ (left) and grid-like points including $X$ (right).
  • Figure 3: Kernels $\phi_{3,3}$, $\phi_8$, $K$ (left to right), and their support (black line).
  • Figure 4: Grid-like data $X_{3,5} = X_3 \times X_5$ (left) and Franke's function (right).
  • Figure 5: Comparison between kernels $\phi_{1,3}$ and $\phi_8$. Condition number of kernel matrices on $X_j$ in (\ref{['component:set']}), $j=1,\ldots,8$ (left); mean square error (right).
  • ...and 3 more figures

Theorems & Definitions (32)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 3.1
  • Theorem 3.2
  • Definition 3.3
  • Theorem 3.4
  • proof
  • Remark 3.5
  • Theorem 4.1
  • ...and 22 more