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Sparse sampling recovery in integral norms on some function classes

V. Temlyakov

TL;DR

The paper addresses sparse sampling recovery in $L_p$ norms ($2\le p<\infty$) for function classes $\mathbf A^r_\beta(\Psi)$ under universal sampling discretization and $u$-term Nikol\'skii inequalities. It develops a synthesis of Lebesgue-type inequalities for the Weak Orthogonal Matching Pursuit (WOMP) with universal discretization, deriving upper and lower bounds for the optimal recovery error $\varrho_m^o(\mathbf A^r_\beta(\Psi),L_p)$ and establishing near-optimal rates (up to polylog factors) for multivariate function classes, including the trigonometric system. Under $NI(2,p,H,u)$ and USD, WOMP yields $L_2$-norm guarantees on the discretized dictionary and corresponding $L_p$-norm recovery bounds, improving previous results by reducing logarithmic factors. The work also proves fundamental lower bounds for sampling discretization and demonstrates that nonlinear sampling recovery via WOMP provides near-optimal rates for a broad class of systems, including $\mathcal T^d$, thereby advancing the theory of sampling discretization and sparse approximation in high dimensions.

Abstract

This paper is a direct followup of the recent author's paper. In this paper we continue to analyze approximation and recovery properties with respect to systems satisfying universal sampling discretization property and a special unconditionality property. In addition we assume that the subspace spanned by our system satisfies some Nikol'skii-type inequalities. We concentrate on recovery with the error measured in the $L_p$ norm for $2\le p<\infty$. We apply a powerful nonlinear approximation method -- the Weak Orthogonal Matching Pursuit (WOMP) also known under the name Weak Orthogonal Greedy Algorithm (WOGA). We establish that the WOMP based on good points for the $L_2$-universal discretization provides good recovery in the $L_p$ norm for $2\le p<\infty$. For our recovery algorithms we obtain both the Lebesgue-type inequalities for individual functions and the error bounds for special classes of multivariate functions. We combine here two deep and powerful techniques -- Lebesgue-type inequalities for the WOMP and theory of the universal sampling dicretization -- in order to obtain new results in sampling recovery.

Sparse sampling recovery in integral norms on some function classes

TL;DR

The paper addresses sparse sampling recovery in norms () for function classes under universal sampling discretization and -term Nikol\'skii inequalities. It develops a synthesis of Lebesgue-type inequalities for the Weak Orthogonal Matching Pursuit (WOMP) with universal discretization, deriving upper and lower bounds for the optimal recovery error and establishing near-optimal rates (up to polylog factors) for multivariate function classes, including the trigonometric system. Under and USD, WOMP yields -norm guarantees on the discretized dictionary and corresponding -norm recovery bounds, improving previous results by reducing logarithmic factors. The work also proves fundamental lower bounds for sampling discretization and demonstrates that nonlinear sampling recovery via WOMP provides near-optimal rates for a broad class of systems, including , thereby advancing the theory of sampling discretization and sparse approximation in high dimensions.

Abstract

This paper is a direct followup of the recent author's paper. In this paper we continue to analyze approximation and recovery properties with respect to systems satisfying universal sampling discretization property and a special unconditionality property. In addition we assume that the subspace spanned by our system satisfies some Nikol'skii-type inequalities. We concentrate on recovery with the error measured in the norm for . We apply a powerful nonlinear approximation method -- the Weak Orthogonal Matching Pursuit (WOMP) also known under the name Weak Orthogonal Greedy Algorithm (WOGA). We establish that the WOMP based on good points for the -universal discretization provides good recovery in the norm for . For our recovery algorithms we obtain both the Lebesgue-type inequalities for individual functions and the error bounds for special classes of multivariate functions. We combine here two deep and powerful techniques -- Lebesgue-type inequalities for the WOMP and theory of the universal sampling dicretization -- in order to obtain new results in sampling recovery.
Paper Structure (6 sections, 15 theorems, 121 equations)

This paper contains 6 sections, 15 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. New results on upper bounds
  4. New results on lower bounds
  5. Optimal recovery on function classes
  6. Discussion

Key Result

Theorem 1.1

Assume that $\Psi$ is a uniformly bounded Riesz system (in the space $L_2(\Omega,\mu)$) satisfying (ub) and Riesz for some constants $0<R_1\leq R_2<\infty$. Let $2<p<\infty$, $\beta \in (0,1]$, and $r>0$. There exist constants $c'=c'(r,\beta,p,R_1,R_2,d)$ and $C'=C'(r,\beta,p,d)$ such that for any $ for any $m$ satisfying

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem 1.1: VT202
  • Theorem 1.2: DT
  • Theorem 2.1: LT
  • Theorem 2.2: DTM2
  • Theorem 3.1
  • proof
  • Theorem 3.2: DTM3
  • Theorem 3.3
  • proof
  • ...and 12 more