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

arXiv: https://arxiv.org/abs/2502.09794 · HTML: https://mlsanity.com/paper/2502.09794/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 56 references that are in the map.
- First posted on arXiv 13 Feb 2025.

## Builds on (most cited first, 14 of 14)

- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Last Layer Re-Training is Sufficient for Robustness to Spurious Correlations](https://mlsanity.com/paper/2204.02937/) (2022)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [On the approximation of functions by tanh neural networks](https://mlsanity.com/paper/2104.08938/) (2021)
- [The gap between theory and practice in function approximation with deep neural networks](https://mlsanity.com/paper/2001.07523/) (2020)
- [Near-Minimax Optimal Estimation With Shallow ReLU Neural Networks](https://mlsanity.com/paper/2109.08844/) (2021)
- [Mathematical theory of deep learning](https://mlsanity.com/paper/2407.18384/) (2024)
- [Deep Neural Networks Are Effective At Learning High-Dimensional Hilbert-Valued Functions From Limited Data](https://mlsanity.com/paper/2012.06081/) (2020)
- [A comparison of deep networks with ReLU activation function and linear spline-type methods](https://mlsanity.com/paper/1804.02253/) (2018)
- [Optimal deep learning of holomorphic operators between Banach spaces](https://mlsanity.com/paper/2406.13928/) (2024)
- [A practical existence theorem for reduced order models based on convolutional autoencoders](https://mlsanity.com/paper/2402.00435/) (2024)
- [Learning smooth functions in high dimensions: from sparse polynomials to deep neural networks](https://mlsanity.com/paper/2404.03761/) (2024)
- [Analysis of Deep Neural Networks with Quasi-optimal polynomial approximation rates](https://mlsanity.com/paper/1912.02302/) (2019)
- [Physics-informed deep learning and compressive collocation for high-dimensional diffusion-reaction equations: practical existence theory and numerics](https://mlsanity.com/paper/2406.01539/) (2024)

## Abstract (arXiv)

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.