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Sampling recovery on classes defined by integral operators and sparse approximation with adaptive dictionaries

V. Temlyakov

TL;DR

The paper addresses the problem of understanding the asymptotics of sampling-recovery errors for collections of function classes ${\mathbf W}^K_q$ defined by integral operators with kernels from a given class, using the asymptotic characteristic $ac_n({\mathbf W}^K_q,L_p)$. It establishes a deep link between sampling recovery and sparse nonlinear approximation with adaptive dictionaries, and derives upper and lower bounds for the recovery errors across regimes of $q$ and $p$, by connecting these problems to adaptive cross-function dictionaries and Kolmogorov-width results. The work defines the function classes ${\mathbf W}^\mathbf a_\mathbf q$ and ${\mathbf H}^\mathbf a_\mathbf q$, introduces adaptive dictionaries including ${\mathcal{L}}{\mathcal{K}}$, ${\mathcal{K}}{\mathcal{L}}$, and related systems, and demonstrates how adaptive sparse approximation informs optimal sampling strategies. Overall, the results broaden the theoretical understanding of sampling schemes for kernel-defined smoothness classes and introduce an adaptive-dictionary framework with potential implications for high-dimensional approximation and numerical integration.

Abstract

In this paper we continue to develop the following general approach. We study asymptotic behavior of the errors of sampling recovery not for an individual smoothness class, how it is usually done, but for the collection of classes, which are defined by integral operators with kernels coming from a given class of functions. Earlier, such approach was realized for the Kolmogorov widths and very recently for the entropy numbers. It turns out that the above problem is closely related to the sparse approximation problem with respect to different redundant dictionaries. Specifically, the problem of sampling recovery is connected with sparse nonlinear approximation with respect to adaptive dictionaries, which means that the dictionary depends on the function under approximation.

Sampling recovery on classes defined by integral operators and sparse approximation with adaptive dictionaries

TL;DR

The paper addresses the problem of understanding the asymptotics of sampling-recovery errors for collections of function classes defined by integral operators with kernels from a given class, using the asymptotic characteristic . It establishes a deep link between sampling recovery and sparse nonlinear approximation with adaptive dictionaries, and derives upper and lower bounds for the recovery errors across regimes of and , by connecting these problems to adaptive cross-function dictionaries and Kolmogorov-width results. The work defines the function classes and , introduces adaptive dictionaries including , , and related systems, and demonstrates how adaptive sparse approximation informs optimal sampling strategies. Overall, the results broaden the theoretical understanding of sampling schemes for kernel-defined smoothness classes and introduce an adaptive-dictionary framework with potential implications for high-dimensional approximation and numerical integration.

Abstract

In this paper we continue to develop the following general approach. We study asymptotic behavior of the errors of sampling recovery not for an individual smoothness class, how it is usually done, but for the collection of classes, which are defined by integral operators with kernels coming from a given class of functions. Earlier, such approach was realized for the Kolmogorov widths and very recently for the entropy numbers. It turns out that the above problem is closely related to the sparse approximation problem with respect to different redundant dictionaries. Specifically, the problem of sampling recovery is connected with sparse nonlinear approximation with respect to adaptive dictionaries, which means that the dictionary depends on the function under approximation.
Paper Structure (10 sections, 21 theorems, 118 equations)

This paper contains 10 sections, 21 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Sampling recovery
  3. Adaptive dictionaries
  4. Function classes
  5. Some known results on sampling recovery
  6. Proofs
  7. Some connections
  8. Upper bounds
  9. Lower bounds
  10. Discussion

Key Result

Theorem 1.1

Let $d=1$ and ${\mathbf F}^\mathbf r_1$ denote one of the classes ${\mathbf W}^\mathbf r_1$ or ${\mathbf H}^\mathbf r_1$ (see the definition in Section fc below) of functions of two variables. Then for $1\le q,p \le \infty$ and $\mathbf r > (1,1+\max(1/2,1/q))$ we have with $\xi(q,p)$ defined in (ksi').

Theorems & Definitions (25)

  • Theorem 1.1: VT32
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9: VT32, Theorem 2
  • Definition 2.1
  • ...and 15 more