# A function space analysis of finite neural networks with insights from sampling theory

arXiv: https://arxiv.org/abs/2004.06989 · HTML: https://mlsanity.com/paper/2004.06989/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Apr 2020.

## Built on by (most cited first, 3 of 3)

- [NeuralMLS: Geometry-Aware Control Point Deformation](https://mlsanity.com/paper/2201.01873/) (2022)
- [Mesh Draping: Parametrization-Free Neural Mesh Transfer](https://mlsanity.com/paper/2110.05433/) (2021)
- [Stretched and measured neural predictions of complex network dynamics](https://mlsanity.com/paper/2301.04900/) (2023)

## Abstract (arXiv)

This work suggests using sampling theory to analyze the function space represented by neural networks. First, it shows, under the assumption of a finite input domain, which is the common case in training neural networks, that the function space generated by multi-layer networks with non-expansive activation functions is smooth. This extends over previous works that show results for the case of infinite width ReLU networks. Then, under the assumption that the input is band-limited, we provide novel error bounds for univariate neural networks. We analyze both deterministic uniform and random sampling showing the advantage of the former.