# Universal Approximation Theorem for a Single-Layer Transformer

arXiv: https://arxiv.org/abs/2507.10581 · HTML: https://mlsanity.com/paper/2507.10581/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 19 references that are in the map.
- First posted on arXiv 11 Jul 2025.

## Builds on (most cited first, 12 of 12)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Sequence to Sequence Learning with Neural Networks](https://mlsanity.com/paper/1409.3215/) (2014)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Are Transformers universal approximators of sequence-to-sequence functions?](https://mlsanity.com/paper/1912.10077/) (2019)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Are Transformers with One Layer Self-Attention Using Low-Rank Weight Matrices Universal Approximators?](https://mlsanity.com/paper/2307.14023/) (2023)

## Built on by (most cited first, 1 of 1)

- [The Role of Feed-Forward Layers in Transformer Dynamics](https://mlsanity.com/paper/2609.36230/) (2026)

## Abstract (arXiv)

Deep learning employs multi-layer neural networks trained via the backpropagation algorithm. This approach has achieved success across many domains and relies on adaptive gradient methods such as the Adam optimizer. Sequence modeling evolved from recurrent neural networks to attention-based models, culminating in the Transformer architecture. Transformers have achieved state-of-the-art performance in natural language processing (for example, BERT and GPT-3) and have been applied in computer vision and computational biology. However, theoretical understanding of these models remains limited. In this paper, we examine the mathematical foundations of deep learning and Transformers and present a novel theoretical result. We review key concepts from linear algebra, probability, and optimization that underpin deep learning, and we analyze the multi-head self-attention mechanism and the backpropagation algorithm in detail. Our main contribution is a universal approximation theorem for Transformers: we prove that a single-layer Transformer, comprising one self-attention layer followed by a position-wise feed-forward network with ReLU activation, can approximate any continuous sequence-to-sequence mapping on a compact domain to arbitrary precision. We provide a formal statement and a complete proof. Finally, we present case studies that demonstrate the practical implications of this result. Our findings advance the theoretical understanding of Transformer models and help bridge the gap between theory and practice.