# A General Method for Proving Networks Universal Approximation Property

arXiv: https://arxiv.org/abs/2511.07857 · HTML: https://mlsanity.com/paper/2511.07857/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 13 references that are in the map.
- First posted on arXiv 11 Nov 2025.

## Builds on (most cited first, 7 of 7)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Fully Convolutional Networks for Semantic Segmentation](https://mlsanity.com/paper/1411.4038/) (2014)
- [Are Transformers universal approximators of sequence-to-sequence functions?](https://mlsanity.com/paper/1912.10077/) (2019)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Approximation and Non-parametric Estimation of ResNet-type Convolutional Neural Networks](https://mlsanity.com/paper/1903.10047/) (2019)

## Abstract (arXiv)

Deep learning architectures are highly diverse. To prove their universal approximation properties, existing works typically rely on model-specific proofs. Generally, they construct a dedicated mathematical formulation for each architecture (e.g., fully connected networks, CNNs, or Transformers) and then prove their universal approximability. However, this approach suffers from two major limitations: first, every newly proposed architecture often requires a completely new proof from scratch; second, these proofs are largely isolated from one another, lacking a common analytical foundation. This not only incurs significant redundancy but also hinders unified theoretical understanding across different network families. To address these issues, this paper proposes a general and modular framework for proving universal approximation. We define a basic building block (comprising one or multiple layers) that possesses the universal approximation property as a Universal Approximation Module (UAM). Under this condition, we show that any deep network composed of such modules inherently retains the universal approximation property. Moreover, the overall approximation process can be interpreted as a progressive refinement across modules. This perspective not only unifies the analysis of diverse architectures but also enables a step-by-step understanding of how expressive power evolves through the network.