The devil in the (de)tails: an improved recovery guarantee for sparse approximation

2026-06-24

AI Summary

This work addresses the issue of excessively large dictionaries and high computational costs in sparse approximation of high-dimensional functions, which arises from overly conservative $L^\infty$ truncation error bounds. By introducing i.i.d. sampling randomness into the truncation error analysis for the first time, the authors derive a sharp discrete $L^2$ error bound that accurately captures the decay behavior of the continuous $L^2$ norm. This approach substantially reduces the size of the truncated index set and yields improved sparse recovery guarantees in both weighted Wiener spaces and anisotropic Sobolev spaces compared to existing results. Additionally, the measurement conditions under bounded Riesz systems are refined to exhibit weaker dependence on the Riesz constants and to possess scale invariance. The theoretical framework integrates tools from compressed sensing, random sampling, function space analysis, and Riesz system theory.

Abstract

Many functions exhibit approximate sparsity in their coefficients with respect to a given dictionary. In recent literature, sparse approximation in such a dictionary from i.i.d. pointwise samples, underpinned by compressed sensing, has become a powerful tool for high-dimensional function approximation. A key step in this framework is truncating the (typically countably-infinite) dictionary to a finite index set of size $n$, so that compressed sensing tools can be used to approximate the function by a sparse combination of these truncated dictionary elements. This introduces a discrete $L^2$-truncation error over the sample points, which in standard approaches, is bounded by the continuous $L^\infty$-norm. Such a deterministic, worst-case bound ignores the randomness of the sample points entirely. As a result, $n$ must be taken unnecessarily large to keep the truncation error under control, which directly inflates the size of the matrix involved in the sparse recovery algorithm and increases computational cost. In this paper, we show that by exploiting the i.i.d. structure of the sample points, the discrete $L^2$ truncation error admits a bound that instead reflects the faster decay behaviour of the continuous $L^2$-norm truncation error and yields significantly smaller truncation sets and decreased computational cost. We demonstrate this through applications to weighted Wiener spaces and anisotropic Sobolev spaces, in each case obtaining significantly smaller truncation sets than recent works. In addition, we also present an improved bound of independent interest for sparse approximation in bounded Riesz systems, where the measurement condition exhibits a smaller (and scale-invariant) dependence on the Riesz constants than in previous works.

Problem

• sparse approximation
• truncation error
• compressed sensing
• random sampling
• high-dimensional function approximation

Innovation

• sparse approximation
• truncation error
• compressed sensing
• Riesz systems
• random sampling