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High-dimensional sparse trigonometric approximation in the uniform norm and consequences for sampling recovery

Moritz Moeller, Serhii Stasyuk, Tino Ullrich

TL;DR

This work advances high-dimensional sparse trigonometric approximation by deriving dimension-controlled bounds for best $m$-term widths in Wiener spaces $\\mathcal{A}_\\theta$ and transferring these results to the uniform norm via an improved Nikol'skij inequality. The authors develop a probabilistic construction to approximate Wiener-class functions in $L_q$ with $2\\le q<\\infty$, achieving rate $\\sigma_{4m}(\\mathcal{A}_\\theta;\\mathcal{T}^d)_{L_q} \\le C \,\\sqrt{q} \, m^{-(1/\\theta-1/2)}$ with an absolute $C<27$, and extend to $L_\\infty$ with a dimension-dependent factor $\\sqrt{d}$ and a logarithmic term. These bounds feed into Besov spaces with mixed smoothness, yielding nonlinear sampling-number estimates that scale polynomially in the inverse accuracy and have controlled $d$-dependence, thereby supporting tractable sampling recovery via $\\ell_1$-minimization. Collectively, the results complement prior polynomial-tractability findings and illuminate the role of sparsity and mixed-smoothness in high-dimensional approximation and sampling.

Abstract

Recent findings by Jahn, T. Ullrich, Voigtlaender [10] relate non-linear sampling numbers for the square norm to quantities involving trigonometric best $m-$term approximation errors in the uniform norm. Here we establish new results for sparse trigonometric approximation with respect to the high-dimensional setting, where the influence of the dimension $d$ has to be controlled. In particular, we focus on best $m-$term trigonometric approximation for (unweighted) Wiener classes in $L_q$ and give precise constants. Our main results are approximation guarantees where the number of terms $m$ scales at most quadratic in the inverse accuracy $1/\varepsilon$. Providing a refined version of the classical Nikol'skij inequality we are able to extrapolate the $L_q$-result to $L_\infty$ while limiting the influence of the dimension to a $\sqrt{d}$-factor and an additonal $\log$-term in the size of the (rectangular) spectrum. This has consequences for the tractable sampling recovery via $\ell_1$-minimization of functions belonging to certain Besov classes with bounded mixed smoothness. This complements polynomial tractability results recently given by Krieg [12].

High-dimensional sparse trigonometric approximation in the uniform norm and consequences for sampling recovery

TL;DR

This work advances high-dimensional sparse trigonometric approximation by deriving dimension-controlled bounds for best -term widths in Wiener spaces and transferring these results to the uniform norm via an improved Nikol'skij inequality. The authors develop a probabilistic construction to approximate Wiener-class functions in with , achieving rate with an absolute , and extend to with a dimension-dependent factor and a logarithmic term. These bounds feed into Besov spaces with mixed smoothness, yielding nonlinear sampling-number estimates that scale polynomially in the inverse accuracy and have controlled -dependence, thereby supporting tractable sampling recovery via -minimization. Collectively, the results complement prior polynomial-tractability findings and illuminate the role of sparsity and mixed-smoothness in high-dimensional approximation and sampling.

Abstract

Recent findings by Jahn, T. Ullrich, Voigtlaender [10] relate non-linear sampling numbers for the square norm to quantities involving trigonometric best term approximation errors in the uniform norm. Here we establish new results for sparse trigonometric approximation with respect to the high-dimensional setting, where the influence of the dimension has to be controlled. In particular, we focus on best term trigonometric approximation for (unweighted) Wiener classes in and give precise constants. Our main results are approximation guarantees where the number of terms scales at most quadratic in the inverse accuracy . Providing a refined version of the classical Nikol'skij inequality we are able to extrapolate the -result to while limiting the influence of the dimension to a -factor and an additonal -term in the size of the (rectangular) spectrum. This has consequences for the tractable sampling recovery via -minimization of functions belonging to certain Besov classes with bounded mixed smoothness. This complements polynomial tractability results recently given by Krieg [12].
Paper Structure (5 sections, 9 theorems, 74 equations)

This paper contains 5 sections, 9 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Trigonometric best $m$-term widths and Wiener spaces
  3. Nikol'skij's inequality revisited
  4. Sparse trigonometric approximation in $L_q$, $2\leq q \leq \infty$
  5. Application -- Besov spaces with mixed smoothness

Key Result

Lemma 2.2

Let $0 < \theta < \gamma \leq \infty$. Then it holds for all $f\in \mathcal{A}_{\theta}$ . As usual, if $\gamma = \infty$ the sum in the middle is replaced by the $\sup$ over $j$.

Theorems & Definitions (19)

  • Definition 2.1
  • Lemma 2.2
  • Theorem 3.1: Nikol'skij's inequality
  • proof
  • Lemma 4.1: Bernstein inequality, FoRa13
  • Corollary 4.2
  • proof
  • Corollary 4.3: Tails to moments
  • proof
  • Theorem 4.4
  • ...and 9 more