# On the Expressiveness of Softmax Attention: A Recurrent Neural Network Perspective

arXiv: https://arxiv.org/abs/2507.23632 · HTML: https://mlsanity.com/paper/2507.23632/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 42 references that are in the map.
- First posted on arXiv 31 Jul 2025.

## Builds on (most cited first, 30 of 38)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [wav2vec 2.0: A Framework for Self-Supervised Learning of Speech Representations](https://mlsanity.com/paper/2006.11477/) (2020)
- [Longformer: The Long-Document Transformer](https://mlsanity.com/paper/2004.05150/) (2020)
- [Sequence to Sequence Learning with Neural Networks](https://mlsanity.com/paper/1409.3215/) (2014)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [A Time Series is Worth 64 Words: Long-term Forecasting with Transformers](https://mlsanity.com/paper/2211.14730/) (2022)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [GLU Variants Improve Transformer](https://mlsanity.com/paper/2002.05202/) (2020)
- [Big Bird: Transformers for Longer Sequences](https://mlsanity.com/paper/2007.14062/) (2020)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [Linformer: Self-Attention with Linear Complexity](https://mlsanity.com/paper/2006.04768/) (2020)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale](https://mlsanity.com/paper/2406.17557/) (2024)
- [Efficient Attention: Attention with Linear Complexities](https://mlsanity.com/paper/1812.01243/) (2018)
- [Learning to (Learn at Test Time): RNNs with Expressive Hidden States](https://mlsanity.com/paper/2407.04620/) (2024)
- [Random Feature Attention](https://mlsanity.com/paper/2103.02143/) (2021)
- [Titans: Learning to Memorize at Test Time](https://mlsanity.com/paper/2501.00663/) (2024)
- [Gated Attention for Large Language Models: Non-linearity, Sparsity, and Attention-Sink-Free](https://mlsanity.com/paper/2505.06708/) (2025)
- [cosFormer: Rethinking Softmax in Attention](https://mlsanity.com/paper/2202.08791/) (2022)
- [Stabilizing Transformer Training by Preventing Attention Entropy Collapse](https://mlsanity.com/paper/2303.06296/) (2023)
- [RWKV-7 "Goose" with Expressive Dynamic State Evolution](https://mlsanity.com/paper/2503.14456/) (2025)
- [SlimPajama-DC: Understanding Data Combinations for LLM Training](https://mlsanity.com/paper/2309.10818/) (2023)

## Built on by (most cited first, 6 of 6)

- [Affine-Scaled Attention: Towards Flexible and Stable Transformer Attention](https://mlsanity.com/paper/2602.23057/) (2026)
- [Energy-Entropy Regularization: The True Power of Minimal Looped Transformers](https://mlsanity.com/paper/2601.09588/) (2026)
- [Don't Read Everything: A Curvature-Conditioned Query for Linear Attention](https://mlsanity.com/paper/2606.01294/) (2026)
- [Softmax Linear Attention: Reclaiming Global Competition](https://mlsanity.com/paper/2602.01744/) (2026)
- [2Mamba2Furious: Linear in Complexity, Competitive in Accuracy](https://mlsanity.com/paper/2602.17363/) (2026)
- [Black-Mamba: Biologically-Inspired Leaky Accumulation for Conceptual Knowledge under Distribution Drift](https://mlsanity.com/paper/2607.18899/) (2026)

## Abstract (arXiv)

Since its introduction, softmax attention has become the backbone of modern transformer architectures due to its expressiveness and scalability across a wide range of tasks. However, the main drawback of softmax attention is the quadratic memory requirement and computational complexity with respect to the sequence length. By replacing the softmax nonlinearity, linear attention and similar methods have been introduced to avoid the quadratic bottleneck of softmax attention. Despite these linear forms of attention being derived from the original softmax formulation, they typically lag in terms of downstream accuracy. While strong intuition of the softmax nonlinearity on the query and key inner product suggests that it has desirable properties compared to other nonlinearities, the question of why this discrepancy exists still remains unanswered. This work demonstrates that linear attention is an approximation of softmax attention by deriving the recurrent form of softmax attention. Using this form, each part of softmax attention can be described in the language of recurrent neural networks (RNNs). Describing softmax attention as an RNN allows for the ablation of the components of softmax attention to understand the importance of each part and how they interact. In this way, our work helps explain why softmax attention is more expressive than its counterparts.