# cosFormer: Rethinking Softmax in Attention

arXiv: https://arxiv.org/abs/2202.08791 · HTML: https://mlsanity.com/paper/2202.08791/

## Key facts

- Cited by 214 papers in the map (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 39 of its 52 references that are in the map.
- First posted on arXiv 17 Feb 2022.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 30 of 39)

- [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)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [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)
- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [End-to-End Object Detection with Transformers](https://mlsanity.com/paper/2005.12872/) (2020)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [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)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Big Bird: Transformers for Longer Sequences](https://mlsanity.com/paper/2007.14062/) (2020)
- [Reformer: The Efficient Transformer](https://mlsanity.com/paper/2001.04451/) (2020)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [Generating Long Sequences with Sparse Transformers](https://mlsanity.com/paper/1904.10509/) (2019)
- [Linformer: Self-Attention with Linear Complexity](https://mlsanity.com/paper/2006.04768/) (2020)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [What Does BERT Look At? An Analysis of BERT's Attention](https://mlsanity.com/paper/1906.04341/) (2019)
- [Long Range Arena: A Benchmark for Efficient Transformers](https://mlsanity.com/paper/2011.04006/) (2020)
- [Efficient Content-Based Sparse Attention with Routing Transformers](https://mlsanity.com/paper/2003.05997/) (2020)
- [Nystr"omformer: A Nystr"om-Based Algorithm for Approximating Self-Attention](https://mlsanity.com/paper/2102.03902/) (2021)
- [Random Feature Attention](https://mlsanity.com/paper/2103.02143/) (2021)
- [Revealing the Dark Secrets of BERT](https://mlsanity.com/paper/1908.08593/) (2019)
- [Sparse Sinkhorn Attention](https://mlsanity.com/paper/2002.11296/) (2020)
- [Adaptive Input Representations for Neural Language Modeling](https://mlsanity.com/paper/1809.10853/) (2018)
- [Transformer Dissection: A Unified Understanding of Transformer's Attention via the Lens of Kernel](https://mlsanity.com/paper/1908.11775/) (2019)

## Built on by (most cited first, 30 of 214)

- [Spikformer: When Spiking Neural Network Meets Transformer](https://mlsanity.com/paper/2209.15425/) (2022)
- [MiniMax-M1: Scaling Test-Time Compute Efficiently with Lightning Attention](https://mlsanity.com/paper/2506.13585/) (2025)
- [Long Range Language Modeling via Gated State Spaces](https://mlsanity.com/paper/2206.13947/) (2022)
- [FLatten Transformer: Vision Transformer using Focused Linear Attention](https://mlsanity.com/paper/2308.00442/) (2023)
- [Spike-driven Transformer](https://mlsanity.com/paper/2307.01694/) (2023)
- [HGRN2: Gated Linear RNNs with State Expansion](https://mlsanity.com/paper/2404.07904/) (2024)
- [The Hedgehog & the Porcupine: Expressive Linear Attentions with Softmax Mimicry](https://mlsanity.com/paper/2402.04347/) (2024)
- [LongCoder: A Long-Range Pre-trained Language Model for Code Completion](https://mlsanity.com/paper/2306.14893/) (2023)
- [Demystify Mamba in Vision: A Linear Attention Perspective](https://mlsanity.com/paper/2405.16605/) (2024)
- [Hierarchically Gated Recurrent Neural Network for Sequence Modeling](https://mlsanity.com/paper/2311.04823/) (2023)
- [The Devil in Linear Transformer](https://mlsanity.com/paper/2210.10340/) (2022)
- [What Makes Convolutional Models Great on Long Sequence Modeling?](https://mlsanity.com/paper/2210.09298/) (2022)
- [KERPLE: Kernelized Relative Positional Embedding for Length Extrapolation](https://mlsanity.com/paper/2205.09921/) (2022)
- [MB-TaylorFormer: Multi-branch Efficient Transformer Expanded by Taylor Formula for Image Dehazing](https://mlsanity.com/paper/2308.14036/) (2023)
- [Beyond the Limits: A Survey of Techniques to Extend the Context Length in Large Language Models](https://mlsanity.com/paper/2402.02244/) (2024)
- [Test-time regression: a unifying framework for designing sequence models with associative memory](https://mlsanity.com/paper/2501.12352/) (2025)
- [Polynormer: Polynomial-Expressive Graph Transformer in Linear Time](https://mlsanity.com/paper/2403.01232/) (2024)
- [Mamba-360: Survey of State Space Models as Transformer Alternative for Long Sequence Modelling: Methods, Applications, and Challenges](https://mlsanity.com/paper/2404.16112/) (2024)
- [Implicit Motion Handling for Video Camouflaged Object Detection](https://mlsanity.com/paper/2203.07363/) (2022)
- [SLA: Beyond Sparsity in Diffusion Transformers via Fine-Tunable Sparse-Linear Attention](https://mlsanity.com/paper/2509.24006/) (2025)
- [Linearizing Large Language Models](https://mlsanity.com/paper/2405.06640/) (2024)
- [Bridging the Divide: Reconsidering Softmax and Linear Attention](https://mlsanity.com/paper/2412.06590/) (2024)
- [EGformer: Equirectangular Geometry-biased Transformer for 360 Depth Estimation](https://mlsanity.com/paper/2304.07803/) (2023)
- [How to Capture Higher-order Correlations? Generalizing Matrix Softmax Attention to Kronecker Computation](https://mlsanity.com/paper/2310.04064/) (2023)
- [Toeplitz Neural Network for Sequence Modeling](https://mlsanity.com/paper/2305.04749/) (2023)
- [Log-Linear Attention](https://mlsanity.com/paper/2506.04761/) (2025)
- [Jet-Nemotron: Efficient Language Model with Post Neural Architecture Search](https://mlsanity.com/paper/2508.15884/) (2025)
- [Scene Adaptive Sparse Transformer for Event-based Object Detection](https://mlsanity.com/paper/2404.01882/) (2024)
- [A Survey of Techniques for Optimizing Transformer Inference](https://mlsanity.com/paper/2307.07982/) (2023)
- [MB-TaylorFormer V2: Improved Multi-branch Linear Transformer Expanded by Taylor Formula for Image Restoration](https://mlsanity.com/paper/2501.04486/) (2025)

## Abstract (arXiv)

Transformer has shown great successes in natural language processing, computer vision, and audio processing. As one of its core components, the softmax attention helps to capture long-range dependencies yet prohibits its scale-up due to the quadratic space and time complexity to the sequence length. Kernel methods are often adopted to reduce the complexity by approximating the softmax operator. Nevertheless, due to the approximation errors, their performances vary in different tasks/corpus and suffer crucial performance drops when compared with the vanilla softmax attention. In this paper, we propose a linear transformer called cosFormer that can achieve comparable or better accuracy to the vanilla transformer in both casual and cross attentions. cosFormer is based on two key properties of softmax attention: i). non-negativeness of the attention matrix; ii). a non-linear re-weighting scheme that can concentrate the distribution of the attention matrix. As its linear substitute, cosFormer fulfills these properties with a linear operator and a cosine-based distance re-weighting mechanism. Extensive experiments on language modeling and text understanding tasks demonstrate the effectiveness of our method. We further examine our method on long sequences and achieve state-of-the-art performance on the Long-Range Arena benchmark. The source code is available at https://github.com/OpenNLPLab/cosFormer.