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Sparse sampling recovery by greedy algorithms

V. Temlyakov

TL;DR

The paper tackles sparse sampling recovery in $L_p$ norms ($1\le p<\infty$) using the Weak Chebyshev Greedy Algorithm (WCGA) for systems with $L_p$-universal sampling discretization and an incoherence property. It develops a discretized WCGA on $\Omega_m$ and proves Lebesgue-type inequalities that tie WCGA performance to best $v$-term approximations, while establishing optimal recovery bounds for multivariate function classes via $\mathbf A^r_\beta(\Psi)$. A generalized RIP framework links universal discretization to stability under sampling, with random sampling yielding high-probability USD in the $1\le p\le 2$ regime. The results demonstrate that nonlinear (sparse) sampling can outperform linear recovery for these classes, provide near-optimal rates (up to logarithmic factors) in $L_p$ norms, and illuminate the practical sampling sizes $m$ needed for reliable recovery. Overall, the work fuses nonlinear greedy approximation with discretization theory to advance sampling recovery in high-dimensional settings.

Abstract

In this paper we analyze approximation and recovery properties with respect to systems satisfying universal sampling discretization property and a special incoherence property. We apply a powerful nonlinear approximation method -- the Weak Chebyshev Greedy Algorithm (WCGA). We establish that the WCGA based on good points for the $L_p$-universal discretization provides good recovery in the $L_p$ norm. For our recovery algorithms we obtain both the Lebesgue-type inequalities for individual functions and the error bounds for special classes of multivariate functions. The main point of the paper is that we combine here two deep and powerful techniques -- Lebesgue-type inequalities for the WCGA and theory of the universal sampling dicretization -- in order to obtain new results in sampling recovery.

Sparse sampling recovery by greedy algorithms

TL;DR

The paper tackles sparse sampling recovery in norms () using the Weak Chebyshev Greedy Algorithm (WCGA) for systems with -universal sampling discretization and an incoherence property. It develops a discretized WCGA on and proves Lebesgue-type inequalities that tie WCGA performance to best -term approximations, while establishing optimal recovery bounds for multivariate function classes via . A generalized RIP framework links universal discretization to stability under sampling, with random sampling yielding high-probability USD in the regime. The results demonstrate that nonlinear (sparse) sampling can outperform linear recovery for these classes, provide near-optimal rates (up to logarithmic factors) in norms, and illuminate the practical sampling sizes needed for reliable recovery. Overall, the work fuses nonlinear greedy approximation with discretization theory to advance sampling recovery in high-dimensional settings.

Abstract

In this paper we analyze approximation and recovery properties with respect to systems satisfying universal sampling discretization property and a special incoherence property. We apply a powerful nonlinear approximation method -- the Weak Chebyshev Greedy Algorithm (WCGA). We establish that the WCGA based on good points for the -universal discretization provides good recovery in the norm. For our recovery algorithms we obtain both the Lebesgue-type inequalities for individual functions and the error bounds for special classes of multivariate functions. The main point of the paper is that we combine here two deep and powerful techniques -- Lebesgue-type inequalities for the WCGA and theory of the universal sampling dicretization -- in order to obtain new results in sampling recovery.
Paper Structure (6 sections, 29 theorems, 145 equations)

This paper contains 6 sections, 29 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Some general results on recovery by WCGA
  3. Applications of Theorem \ref{['CGT2']}
  4. Optimal recovery on function classes
  5. A generalization of the RIP
  6. Discussion

Key Result

Theorem 2.1

Let $X$ be a Banach space satisfying that $\eta(X, w)\le \gamma w^q$, $w>0$ for some parameter $1<q\le 2$. Suppose that ${\mathcal{D}}\subset X$ is a system in $X$ with the ($v,S$)-incoherence property for some integers $1\leq v\leq S$ and parameters $V>0$ and $r>0$. Then the WCGA with weakness para for any positive integer $v$ satisfying $v+u \le S,$ where and $C>1$ is an absolute constant.

Theorems & Definitions (51)

  • Definition 1.1
  • Definition 1.2
  • Remark 2.1
  • Theorem 2.1: VT144, VTbookMA
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • ...and 41 more