Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the optimality of target-data-dependent kernel greedy interpolation in Sobolev Reproducing Kernel Hilbert Spaces

Gabriele Santin, Tizian Wenzel, Bernard Haasdonk

TL;DR

The paper develops a refined convergence theory for target-data-dependent kernel greedy interpolation in Sobolev RKHS by leveraging Li’s entropy-number framework. It shows how weak and adaptive greedy strategies (including $P$-greedy and $f$-greedy) attain algebraic convergence rates tied to entropy decay, with Sobolev-smooth kernels removing prior log-factors and matching optimal nonlinear approximation bounds. The results quantify how adaptivity reduces the interpolation error relative to nonadaptive schemes and establish lower bounds via kernel-matrix eigenvalue analysis, linking performance to the smoothness of the native space. Numerical experiments with Brownian-Bridge kernels corroborate the theoretical rates and illustrate the practical impact of adaptivity on sampling and interpolation efficiency.

Abstract

Kernel interpolation is a versatile tool for the approximation of functions from data, and it can be proven to have some optimality properties when used with kernels related to certain Sobolev spaces. In the context of interpolation, the selection of optimal function sampling locations is a central problem, both from a practical perspective, and as an interesting theoretical question. Greedy interpolation algorithms provide a viable solution for this task, being efficient to run and provably accurate in their approximation. In this paper we close a gap that is present in the convergence theory for these algorithms by employing a recent result on general greedy algorithms. This modification leads to new convergence rates which match the optimal ones when restricted to the $P$-greedy target-data-independent selection rule, and can additionally be proven to be optimal when they fully exploit adaptivity ($f$-greedy). Other than closing this gap, the new results have some significance in the broader setting of the optimality of general approximation algorithms in Reproducing Kernel Hilbert Spaces, as they allow us to compare adaptive interpolation with non-adaptive best nonlinear approximation.

On the optimality of target-data-dependent kernel greedy interpolation in Sobolev Reproducing Kernel Hilbert Spaces

TL;DR

The paper develops a refined convergence theory for target-data-dependent kernel greedy interpolation in Sobolev RKHS by leveraging Li’s entropy-number framework. It shows how weak and adaptive greedy strategies (including -greedy and -greedy) attain algebraic convergence rates tied to entropy decay, with Sobolev-smooth kernels removing prior log-factors and matching optimal nonlinear approximation bounds. The results quantify how adaptivity reduces the interpolation error relative to nonadaptive schemes and establish lower bounds via kernel-matrix eigenvalue analysis, linking performance to the smoothness of the native space. Numerical experiments with Brownian-Bridge kernels corroborate the theoretical rates and illustrate the practical impact of adaptivity on sampling and interpolation efficiency.

Abstract

Kernel interpolation is a versatile tool for the approximation of functions from data, and it can be proven to have some optimality properties when used with kernels related to certain Sobolev spaces. In the context of interpolation, the selection of optimal function sampling locations is a central problem, both from a practical perspective, and as an interesting theoretical question. Greedy interpolation algorithms provide a viable solution for this task, being efficient to run and provably accurate in their approximation. In this paper we close a gap that is present in the convergence theory for these algorithms by employing a recent result on general greedy algorithms. This modification leads to new convergence rates which match the optimal ones when restricted to the -greedy target-data-independent selection rule, and can additionally be proven to be optimal when they fully exploit adaptivity (-greedy). Other than closing this gap, the new results have some significance in the broader setting of the optimality of general approximation algorithms in Reproducing Kernel Hilbert Spaces, as they allow us to compare adaptive interpolation with non-adaptive best nonlinear approximation.
Paper Structure (15 sections, 11 theorems, 97 equations, 4 figures)

This paper contains 15 sections, 11 theorems, 97 equations, 4 figures.

Table of Contents

  1. Interpolation and greedy algorithms
  2. A general result on entropy numbers in Hilbert spaces
  3. A refined convergence theory for greedy kernel interpolation
  4. Weak greedy algorithms
  5. Preliminaries
  6. Convergence in terms of entropy numbers
  7. Entropy numbers of the kernel dictionary and algebraic error rates
  8. Weakened convergence for weaker selections
  9. Discussion of the results and of their optimality
  10. Optimality of some components of the estimates
  11. Comparison with optimal uniform nonlinear approximation
  12. A limiting case for beta equal 1
  13. Numerical experiments
  14. Additional proofs
  15. Calculations for the example in Section \ref{['sec:bb_example']}

Key Result

Lemma 1

Let $K\subset\mathcal{H}$ be a precompact set in a Hilbert space $\mathcal{H}$, let $n\in\mathbb{N}, m\in\mathbb{N}_0$, and let $v_1,\dots, v_{n+m}\in K$ be arbitrary. Define $V_0\coloneqq\{0\}$, $V_i\coloneqq\mathop{\mathrm{span}}\nolimits\{v_j, 1\leq j\leq i\}$. Then where $S_n\coloneqq \pi^{n/2} / \Gamma(n/2+1)$ is the volume of the unit ball in $\mathbb{R}^n$.

Figures (4)

  • Figure 1: Visualization of the new rates of algebraic convergence of the $\beta$-greedy algorithms when $\mathcal{H}_k(\Omega)$ is continuously embedded in $W^\tau_2(\Omega)$, $\Omega\subset\mathbb{R}^d$, $\tau>d/2$. The lines show the rates for $d=2$ and $\tau_1=1$ (i.e., $\tau_1/d-1/2=0$ -- shown to illustrate the limiting case), $\tau_2=3/2$ (i.e., $\tau_2/d-1/2\leq1/2$), and $\tau_3=3$ (i.e., $\tau_3/d-1/2>1/2$).
  • Figure 2: Decay of the error ($y$-axis) for $f\coloneqq x(1-x)$ as a function of the number of interpolation points ($x$-axis), interpolated with the Brownian Bridge kernel in $\Omega=(0,1)$ by $f$-greedy and optimal points. The plots show the decay of the $L_\infty$-error (left), the $\mathcal{H}_k(\Omega)$-error (right), and their ratio (bottom), together with their algebraic rates. The $x$- and $y$-scales of the plot are logarithmic.
  • Figure 3: Results of the interpolation of $f_p$, for values $p=0.51, 3$, by $f$-greedy in $\Omega=(0,1)$ with kernel $k_1$. The plots show the decay of the $L_\infty$-error (top left), the $\mathcal{H}_k(\Omega)$-error (top right), and their ratio (bottom left), together with their estimated algebraic rates. The bottom right figure shows a density plot of the selected interpolation points, i.e., a count of the number of points selected in each represented subinterval.
  • Figure 4: Results of the interpolation of $f_p$, for values $p=1.51, 4$, by $f$-greedy in $\Omega=(0,1)$ with kernel $k_2$. The plots show the decay of the $L_\infty$-error (top left), the $\mathcal{H}_k(\Omega)$-error (top right), and their ratio (bottom left), together with their estimated algebraic rates. The bottom right figure shows a density plot of the selected interpolation points, i.e., a count of the number of points selected in each represented subinterval.

Theorems & Definitions (27)

  • Lemma 1
  • proof
  • Remark 2
  • Remark 3
  • Lemma 4
  • Remark 5
  • Proposition 6
  • proof
  • Theorem 7
  • proof
  • ...and 17 more