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On the convergence of generalized kernel-based interpolation by greedy data selection algorithms

Kristof Albrecht, Armin Iske

TL;DR

This work analyzes the convergence of generalized kernel-based interpolation methods and proves convergence of popular greedy data selection algorithms for totally bounded sets of sampling functionals for totally bounded sets of sampling functionals.

Abstract

We analyze the convergence of generalized kernel-based interpolation methods. This is done under minimalistic assumptions on both the kernel and the target function. On these grounds, we further prove convergence of popular greedy data selection algorithms for totally bounded sets of sampling functionals. Supporting numerical results concerning computerized tomography are provided for illustration.

On the convergence of generalized kernel-based interpolation by greedy data selection algorithms

TL;DR

This work analyzes the convergence of generalized kernel-based interpolation methods and proves convergence of popular greedy data selection algorithms for totally bounded sets of sampling functionals for totally bounded sets of sampling functionals.

Abstract

We analyze the convergence of generalized kernel-based interpolation methods. This is done under minimalistic assumptions on both the kernel and the target function. On these grounds, we further prove convergence of popular greedy data selection algorithms for totally bounded sets of sampling functionals. Supporting numerical results concerning computerized tomography are provided for illustration.
Paper Structure (9 sections, 9 theorems, 63 equations, 4 figures, 1 table)

This paper contains 9 sections, 9 theorems, 63 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Generalized kernel-based interpolation
  3. Convergence under mild conditions
  4. Parameterization
  5. Greedy data selection algorithms
  6. $\beta$-greedy algorithms
  7. Geometric greedy algorithm
  8. Image reconstruction from Radon data
  9. Acknowledgment

Key Result

Proposition 2.1

Let $\lambda \in \mathcal{H}_K^*$. For the function we have $\lambda^y K(\cdot,y) \in \mathcal{H}_K$. Moreover, the generalized reproduction holds, i.e.,

Figures (4)

  • Figure 1: Reconstruction of Shepp-Logan phantom for different greedy methods after $M = 2,500$ iterations.
  • Figure 2: Reconstruction of the Shepp-Logan phantom. Decay of RMSE (left) and growth of spectral condition number $\kappa_2(A_{K,\Lambda_n})$ (right) as a function of $n$, i.e., the number of the selected Radon functionals.
  • Figure 3: Reconstruction of smooth phantom with smothness parameter $\nu = 3$ for five different greedy methods after $M = 2,500$ iterations.
  • Figure 4: Reconstruction of the smooth phantom. Decay of RMSE (left) and growth of spectral condition number $\kappa_2(A_{K,\Lambda_n})$ (right) as a function of $n$, i.e., the number of the selected Radon functionals.

Theorems & Definitions (23)

  • Proposition 2.1: Wendland2005
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Remark 2.6
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 13 more