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Universal $L_2$-approximation using median lattice algorithms

Zexin Pan, Takashi Goda, Peter Kritzer

TL;DR

The paper introduces a universal median lattice-based approach for $L_2$-approximation in weighted Korobov spaces, eliminating the need for prior knowledge of the smoothness $\alpha$ and weights $\boldsymbol{\gamma}$. By aggregating $R$ randomized rank-1 lattice realizations through medians and adaptively selecting an enlarged hyperbolic cross of Fourier frequencies, the method achieves a probabilistic $L_2$-approximation error with rate approaching the optimal $M^{-\ ext{approx} }$ for the number of evaluations $M$, up to an arbitrarily small $\varepsilon>0$, i.e., near $M^{-\\alpha}$. The authors provide a detailed error decomposition and probabilistic bounds demonstrating near-optimal convergence and discuss tractability under downward-closed and product weights, showing polynomial tractability under suitable summability conditions while noting a linear-in-$d$ dependence of the replication parameter $R$ due to universality. Numerical experiments in a low-dimensional setting corroborate universality and adaptive Fourier coefficient selection, with observed truncation errors closely matching oracle performance and convergence rates consistent with finite-sample behavior. Overall, the work advances universal, parameter-free $L_2$-approximation in high-dimensional settings using median lattice methods, offering practical trade-offs between universality and computational budget.

Abstract

We study the problem of multivariate $L_2$-approximation of functions in a weighted Korobov space using a median lattice-based algorithm recently proposed by the authors. In the original work, the algorithm requires knowledge of the smoothness and weights of the Korobov space to construct the hyperbolic cross index set, where each coefficient is estimated via the median of approximations obtained from randomly shifted, randomly chosen rank-1 lattice rules. In this paper, we introduce a \emph{universal median lattice-based algorithm}, which eliminates the need for any prior information on smoothness and weights. Although the tractability property of the algorithm slightly deteriorates, we prove that, for individual functions in the Korobov space with arbitrary smoothness and (downward-closed) weights, it achieves an $L_2$-approximation error arbitrarily close to the optimal rate with respect to the number of function evaluations. Numerical experiments are conducted to support our theoretical claim.

Universal $L_2$-approximation using median lattice algorithms

TL;DR

The paper introduces a universal median lattice-based approach for -approximation in weighted Korobov spaces, eliminating the need for prior knowledge of the smoothness and weights . By aggregating randomized rank-1 lattice realizations through medians and adaptively selecting an enlarged hyperbolic cross of Fourier frequencies, the method achieves a probabilistic -approximation error with rate approaching the optimal for the number of evaluations , up to an arbitrarily small , i.e., near . The authors provide a detailed error decomposition and probabilistic bounds demonstrating near-optimal convergence and discuss tractability under downward-closed and product weights, showing polynomial tractability under suitable summability conditions while noting a linear-in- dependence of the replication parameter due to universality. Numerical experiments in a low-dimensional setting corroborate universality and adaptive Fourier coefficient selection, with observed truncation errors closely matching oracle performance and convergence rates consistent with finite-sample behavior. Overall, the work advances universal, parameter-free -approximation in high-dimensional settings using median lattice methods, offering practical trade-offs between universality and computational budget.

Abstract

We study the problem of multivariate -approximation of functions in a weighted Korobov space using a median lattice-based algorithm recently proposed by the authors. In the original work, the algorithm requires knowledge of the smoothness and weights of the Korobov space to construct the hyperbolic cross index set, where each coefficient is estimated via the median of approximations obtained from randomly shifted, randomly chosen rank-1 lattice rules. In this paper, we introduce a \emph{universal median lattice-based algorithm}, which eliminates the need for any prior information on smoothness and weights. Although the tractability property of the algorithm slightly deteriorates, we prove that, for individual functions in the Korobov space with arbitrary smoothness and (downward-closed) weights, it achieves an -approximation error arbitrarily close to the optimal rate with respect to the number of function evaluations. Numerical experiments are conducted to support our theoretical claim.

Paper Structure

This paper contains 15 sections, 20 theorems, 122 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. The weighted Korobov space
  4. $L_2$-approximation
  5. Rank-1 lattice rules
  6. The median algorithm and its error
  7. The universal median lattice-based algorithm
  8. Error analysis: the main result
  9. Tractability analysis
  10. Error analysis: bounds on the error decomposition
  11. The first term of \ref{['eqn:errordecom']}
  12. The second term of \ref{['eqn:errordecom']}
  13. The third term of \ref{['eqn:errordecom']}
  14. Numerical experiments
  15. Discussion

Key Result

Lemma 3.4

Let $\boldsymbol{\gamma}=(\gamma_{\mathfrak{u}})_{\mathfrak{u}\subseteq \{1{:}d\}}$ be arbitrary positive weights. Then, for $\alpha>1/2$ and $\lambda > 1/(2\alpha)$, where and $\zeta(q):=\sum_{n=1}^\infty n^{-q}$ denotes the Riemann zeta function.

Figures (2)

  • Figure 1: $L_2$-approximation error for the test functions $f_1$ (left) and $f_2$ (right) as a function of the total number of function evaluations $M$.
  • Figure 2: Normalized truncation error for the test functions $f_1$ (left) and $f_2$ (right). The error from the adaptive selection of the index set is compared to the optimal (oracle) error.

Theorems & Definitions (39)

  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • Corollary 3.7
  • proof
  • Theorem 3.8
  • ...and 29 more