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

2025-02-13

AI Summary

This paper addresses the fundamental signal processing problem of reconstructing frequency-localized (e.g., bandlimited) functions from discrete samples. From an approximation-theoretic perspective, it unifies the analysis of two reconstruction paradigms: least-squares estimation using the Slepian orthogonal basis and deep learning via ReLU neural networks. Theoretical contributions include: (i) the first explicit theorem characterizing the bandwidth–sampling complexity trade-off for least-squares reconstruction in the Slepian basis; and (ii) the first practical existence guarantee for deep learning–based bandlimited function reconstruction, precisely quantifying the interplay among network width, training sample size, and sampling density. Numerical experiments on 1D and 2D tasks demonstrate that deep learning achieves performance comparable to—or even surpassing—that of least-squares methods. The work bridges the typical gap between theoretical feasibility and empirical performance, establishing a new data-driven paradigm for frequency-domain reconstruction.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• Reconstruct frequency-localized functions from samples
• Compare least squares and deep learning methods
• Analyze theoretical and practical approximation challenges

Innovation

• Uses Slepian basis for least squares
• Deep learning for bandlimited functions
• Compares least squares and deep learning