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Error bounds for function approximation using generated sets

Ronald Cools, Dirk Nuyens, Laurence Wilkes

TL;DR

The paper tackles high-dimensional function approximation in reproducing kernel Hilbert spaces with absolutely convergent Fourier series using generated point sets. It develops a probabilistic argument showing that a large class of generators yields worst-case $L_2$-error bounds for the least-squares estimator that match the optimal polynomial convergence rate when the Fourier-weights decay polynomially, and extends these guarantees to rational generators. The results are specialized to the weighted Korobov space, where explicit bounds are derived via spectral decay rates and hyperbolic-cross geometry, including tractability under standard weight-decay conditions. Collectively, the work provides practical guidance for sampling strategy design (both continuous and rational generators) and establishes rigorous convergence guarantees for function approximation in high dimensions with limited samples. These findings have implications for numerical integration and reconstruction tasks in multivariate settings where Fourier-based representations are natural.

Abstract

This paper explores the use of "generated sets" $\{ \{ k \boldsymbolζ \} : k = 1, \ldots, n \}$ for function approximation in reproducing kernel Hilbert spaces which consist of multi-dimensional functions with an absolutely convergent Fourier series. The algorithm is a least squares algorithm that samples the function at the points of a generated set. We show that there exist $\boldsymbolζ \in [0,1]^d$ for which the worst-case $L_2$ error has the optimal order of convergence if the space has polynomially converging approximation numbers. In fact, this holds for a significant portion of the generators. Additionally we show that a restriction to rational generators is possible with a slight increase of the bound. Furthermore, we specialise the results to the weighted Korobov space, where we derive a bound applicable to low values of sample points, and state tractability results.

Error bounds for function approximation using generated sets

TL;DR

The paper tackles high-dimensional function approximation in reproducing kernel Hilbert spaces with absolutely convergent Fourier series using generated point sets. It develops a probabilistic argument showing that a large class of generators yields worst-case -error bounds for the least-squares estimator that match the optimal polynomial convergence rate when the Fourier-weights decay polynomially, and extends these guarantees to rational generators. The results are specialized to the weighted Korobov space, where explicit bounds are derived via spectral decay rates and hyperbolic-cross geometry, including tractability under standard weight-decay conditions. Collectively, the work provides practical guidance for sampling strategy design (both continuous and rational generators) and establishes rigorous convergence guarantees for function approximation in high dimensions with limited samples. These findings have implications for numerical integration and reconstruction tasks in multivariate settings where Fourier-based representations are natural.

Abstract

This paper explores the use of "generated sets" for function approximation in reproducing kernel Hilbert spaces which consist of multi-dimensional functions with an absolutely convergent Fourier series. The algorithm is a least squares algorithm that samples the function at the points of a generated set. We show that there exist for which the worst-case error has the optimal order of convergence if the space has polynomially converging approximation numbers. In fact, this holds for a significant portion of the generators. Additionally we show that a restriction to rational generators is possible with a slight increase of the bound. Furthermore, we specialise the results to the weighted Korobov space, where we derive a bound applicable to low values of sample points, and state tractability results.
Paper Structure (10 sections, 10 theorems, 107 equations, 1 figure)

This paper contains 10 sections, 10 theorems, 107 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and preliminary results
  3. Function space
  4. The Korobov space
  5. Generated sets
  6. The least squares algorithm
  7. Error quantity
  8. The bound for a continuous generator
  9. Generated sets with rational generators
  10. Bounds specific to the weighted Korobov space

Key Result

Lemma 1

The unweighted hyperbolic cross, with $M \in {\mathbb{R}}_{\ge0}$, has cardinality $|A_d(M)| = 0$ for $0 \le M < 1$, $|A_d(M)| = 3^d$ for $1 \le M < 2$, and where the implied constant depends on $d$.

Figures (1)

  • Figure 1: Generated sets with generator ${\boldsymbol{\zeta}} = (\sqrt{2} - 1, \sqrt{3} - 1)$ and progressively more sample points.

Theorems & Definitions (21)

  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Lemma 3
  • proof
  • Corollary 1
  • ...and 11 more