# Clustering in pure-attention hardmax transformers and its role in sentiment analysis

arXiv: https://arxiv.org/abs/2407.01602 · HTML: https://mlsanity.com/paper/2407.01602/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 46 references that are in the map.
- First posted on arXiv 26 Jun 2024.

## Builds on (most cited first, 28 of 28)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Robust Speech Recognition via Large-Scale Weak Supervision](https://mlsanity.com/paper/2212.04356/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Taming Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2012.09841/) (2020)
- [ALBERT: A Lite BERT for Self-supervised Learning of Language Representations](https://mlsanity.com/paper/1909.11942/) (2019)
- [Are Transformers universal approximators of sequence-to-sequence functions?](https://mlsanity.com/paper/1912.10077/) (2019)
- [Generative Language Modeling for Automated Theorem Proving](https://mlsanity.com/paper/2009.03393/) (2020)
- [Molecular Transformer - A Model for Uncertainty-Calibrated Chemical Reaction Prediction](https://mlsanity.com/paper/1811.02633/) (2018)
- [A mathematical perspective on Transformers](https://mlsanity.com/paper/2312.10794/) (2023)
- [The emergence of clusters in self-attention dynamics](https://mlsanity.com/paper/2305.05465/) (2023)
- [Sinkformers: Transformers with Doubly Stochastic Attention](https://mlsanity.com/paper/2110.11773/) (2021)
- [Understanding and Improving Transformer From a Multi-Particle Dynamic System Point of View](https://mlsanity.com/paper/1906.02762/) (2019)
- [Deep Learning via Dynamical Systems: An Approximation Perspective](https://mlsanity.com/paper/1912.10382/) (2019)
- [Neural ODE control for classification, approximation and transport](https://mlsanity.com/paper/2104.05278/) (2021)
- [Sumformer: Universal Approximation for Efficient Transformers](https://mlsanity.com/paper/2307.02301/) (2023)
- [Effects of padding on LSTMs and CNNs](https://mlsanity.com/paper/1903.07288/) (2019)
- [On the infinite-depth limit of finite-width neural networks](https://mlsanity.com/paper/2210.00688/) (2022)
- [Infinitely deep neural networks as diffusion processes](https://mlsanity.com/paper/1905.11065/) (2019)
- [Turnpike in optimal control of PDEs, ResNets, and beyond](https://mlsanity.com/paper/2202.04097/) (2022)
- [Doubly infinite residual neural networks: a diffusion process approach](https://mlsanity.com/paper/2007.03253/) (2020)
- [A deep language model to predict metabolic network equilibria](https://mlsanity.com/paper/2112.03588/) (2021)

## Built on by (most cited first, 15 of 15)

- [Dynamic metastability in the self-attention model](https://mlsanity.com/paper/2410.06833/) (2024)
- [The Mean-Field Dynamics of Transformers](https://mlsanity.com/paper/2512.01868/) (2025)
- [Measure-to-measure interpolation using Transformers](https://mlsanity.com/paper/2411.04551/) (2024)
- [A multiscale analysis of mean-field transformers in the moderate interaction regime](https://mlsanity.com/paper/2509.25040/) (2025)
- [Perceptrons and localization of attention's mean-field landscape](https://mlsanity.com/paper/2601.21366/) (2026)
- [Attention-Only Transformers via Unrolled Subspace Denoising](https://mlsanity.com/paper/2506.03790/) (2025)
- [Random Quadratic Form on a Sphere: Synchronization by Common Noise](https://mlsanity.com/paper/2603.06187/) (2026)
- [Interior interpretability with attention rollout: contraction and propagation profiles in Transformers](https://mlsanity.com/paper/2607.22367/) (2026)
- [Exact Sequence Interpolation with Transformers](https://mlsanity.com/paper/2502.02270/) (2025)
- [Localmax dynamics for attention in transformers and its asymptotic behavior](https://mlsanity.com/paper/2509.15958/) (2025)
- [Dynamical Properties of Tokens in Self-Attention and Effects of Positional Encoding](https://mlsanity.com/paper/2512.03058/) (2025)
- [Control, Optimal Transport and Neural Differential Equations in Supervised Learning](https://mlsanity.com/paper/2503.15105/) (2025)
- [Reachability and asymptotics of Gaussian Transformer dynamics](https://mlsanity.com/paper/2606.07600/) (2026)
- [Machine Learning and Control: Foundations, Advances, and Perspectives](https://mlsanity.com/paper/2510.03303/) (2025)
- [The Attention Within: Consensus Dynamics in Selective State Space Models](https://mlsanity.com/paper/2609.17997/) (2026)

## Abstract (arXiv)

Transformers are extremely successful machine learning models whose mathematical properties remain poorly understood. Here, we rigorously characterize the behavior of transformers with hardmax self-attention and normalization sublayers as the number of layers tends to infinity. By viewing such transformers as discrete-time dynamical systems describing the evolution of points in a Euclidean space, and thanks to a geometric interpretation of the self-attention mechanism based on hyperplane separation, we show that the transformer inputs asymptotically converge to a clustered equilibrium determined by special points called textitleaders. We then leverage this theoretical understanding to solve sentiment analysis problems from language processing using a fully interpretable transformer model, which effectively captures `context' by clustering meaningless words around leader words carrying the most meaning. Finally, we outline remaining challenges to bridge the gap between the mathematical analysis of transformers and their real-life implementation.