Reconstruction of frequency-localized functions from pointwise samples via least squares and deep learning

TL;DR

This paper investigates the recovery of frequency-localized functions from pointwise samples using least squares and deep learning, providing theoretical guarantees and numerical comparisons.

Contribution

It introduces a novel recovery theorem for least squares with the Slepian basis and offers a deep learning approximation guarantee from pointwise data.

Findings

Theoretical recovery guarantees for least squares and deep learning methods.

Numerical comparisons between least squares and deep learning approaches.

Discussion of theoretical limitations and practical implementation gaps.

Abstract

Recovering frequency-localized functions from pointwise data is a fundamental task in signal processing. We examine this problem from an approximation-theoretic perspective, focusing on least squares and deep learning-based methods. First, we establish a novel recovery theorem for least squares approximations using the Slepian basis from uniform random samples in low dimensions, explicitly tracking the dependence of the bandwidth on the sampling complexity. Building on these results, we then present a recovery guarantee for approximating bandlimited functions via deep learning from pointwise data. This result, framed as a practical existence theorem, provides conditions on the network architecture, training procedure, and data acquisition sufficient for accurate approximation. To complement our theoretical findings, we perform numerical comparisons between least squares and deep learning for approximating one- and two-dimensional functions. We conclude with a discussion of the theoretical limitations and the practical gaps between theory and implementation.