# Clustering in Causal Attention Masking

arXiv: https://arxiv.org/abs/2411.04990 · HTML: https://mlsanity.com/paper/2411.04990/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 38 references that are in the map.
- First posted on arXiv 7 Nov 2024.

## Builds on (most cited first, 21 of 21)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [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)
- [Root Mean Square Layer Normalization](https://mlsanity.com/paper/1910.07467/) (2019)
- [Scaling Autoregressive Models for Content-Rich Text-to-Image Generation](https://mlsanity.com/paper/2206.10789/) (2022)
- [Chameleon: Mixed-Modal Early-Fusion Foundation Models](https://mlsanity.com/paper/2405.09818/) (2024)
- [Muse: Text-To-Image Generation via Masked Generative Transformers](https://mlsanity.com/paper/2301.00704/) (2023)
- [Attention is Not All You Need: Pure Attention Loses Rank Doubly Exponentially with Depth](https://mlsanity.com/paper/2103.03404/) (2021)
- [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)
- [Demystifying Oversmoothing in Attention-Based Graph Neural Networks](https://mlsanity.com/paper/2305.16102/) (2023)
- [Dynamic metastability in the self-attention model](https://mlsanity.com/paper/2410.06833/) (2024)
- [How Smooth Is Attention?](https://mlsanity.com/paper/2312.14820/) (2023)
- [Emergence of meta-stable clustering in mean-field transformer models](https://mlsanity.com/paper/2410.23228/) (2024)
- [Geometric Dynamics of Signal Propagation Predict Trainability of Transformers](https://mlsanity.com/paper/2403.02579/) (2024)
- [Synchronization on circles and spheres with nonlinear interactions](https://mlsanity.com/paper/2405.18273/) (2024)
- [On the Regularity of Attention](https://mlsanity.com/paper/2102.05628/) (2021)
- [Generic controllability of equivariant systems and applications to particle systems and neural networks](https://mlsanity.com/paper/2404.08289/) (2024)
- [Expander graphs are globally synchronizing](https://mlsanity.com/paper/2210.12788/) (2022)

## Built on by (most cited first, 30 of 32)

- [The Mean-Field Dynamics of Transformers](https://mlsanity.com/paper/2512.01868/) (2025)
- [Quantitative Clustering in Mean-Field Transformer Models](https://mlsanity.com/paper/2504.14697/) (2025)
- [A multiscale analysis of mean-field transformers in the moderate interaction regime](https://mlsanity.com/paper/2509.25040/) (2025)
- [Critical attention scaling in long-context transformers](https://mlsanity.com/paper/2510.05554/) (2025)
- [The Asymptotic Behavior of Attention in Transformers](https://mlsanity.com/paper/2412.02682/) (2024)
- [Synchronization of mean-field models on the circle](https://mlsanity.com/paper/2507.22857/) (2025)
- [On the Structure of Stationary Solutions to McKean-Vlasov Equations with Applications to Noisy Transformers](https://mlsanity.com/paper/2510.20094/) (2025)
- [Normalization in Attention Dynamics](https://mlsanity.com/paper/2510.22026/) (2025)
- [Optimal Control for Transformer Architectures: Enhancing Generalization, Robustness and Efficiency](https://mlsanity.com/paper/2505.13499/) (2025)
- [Perceptrons and localization of attention's mean-field landscape](https://mlsanity.com/paper/2601.21366/) (2026)
- [Clustering in Deep Stochastic Transformers](https://mlsanity.com/paper/2601.21942/) (2026)
- [OT-Transformer: A Continuous-time Transformer Architecture with Optimal Transport Regularization](https://mlsanity.com/paper/2501.18793/) (2025)
- [Recurrent Self-Attention Dynamics: An Energy-Agnostic Perspective from Jacobians](https://mlsanity.com/paper/2505.19458/) (2025)
- [Quantifying Concentration Phenomena of Mean-Field Transformers in the Low-Temperature Regime](https://mlsanity.com/paper/2605.10931/) (2026)
- [On the number of modes of Gaussian kernel density estimators](https://mlsanity.com/paper/2412.09080/) (2024)
- [Token Sample Complexity of Attention](https://mlsanity.com/paper/2512.10656/) (2025)
- [Exact Sequence Interpolation with Transformers](https://mlsanity.com/paper/2502.02270/) (2025)
- [Gradient Flow Structure and Quantitative Dynamics of Multi-Head Self-Attention](https://mlsanity.com/paper/2605.04279/) (2026)
- [Diffusion Operator Geometry of Feedforward Representations](https://mlsanity.com/paper/2605.01107/) (2026)
- [Training-Induced Escape from Token Clustering in a Mean-Field Formulation of Transformers](https://mlsanity.com/paper/2605.07772/) (2026)
- [Sink vs. diagonal patterns as mechanisms for attention switch and oversmoothing prevention](https://mlsanity.com/paper/2605.08453/) (2026)
- [Multi-Headed Transformer Architectures as Time-dependent Wasserstein Gradient Flows](https://mlsanity.com/paper/2605.18870/) (2026)
- [Attention as Frustrated Synchronization](https://mlsanity.com/paper/2606.18694/) (2026)
- [Feed-Forward Steering in Transformer Residual Dynamics](https://mlsanity.com/paper/2608.02071/) (2026)
- [Faster Query-Key Learning Sharpens Attention in Self-Attention Models](https://mlsanity.com/paper/2608.06776/) (2026)
- [Clustered Attractor Manifolds and Dynamical Condensation in Self-Attention](https://mlsanity.com/paper/2608.08922/) (2026)
- [Multispin Physics of AI Tipping Points and Hallucinations](https://mlsanity.com/paper/2508.01097/) (2025)
- [Kinetic theory for Transformers and the lost-in-the-middle phenomenon](https://mlsanity.com/paper/2605.09213/) (2026)
- [A Unified Framework for Critical Scaling of Inverse Temperature in Self-Attention](https://mlsanity.com/paper/2605.12697/) (2026)
- [Self-Attention Dynamics with Rotary Position Embeddings: Twisted States and Explicit Consensus Rates on the Sphere](https://mlsanity.com/paper/2607.24502/) (2026)

## Abstract (arXiv)

This work presents a modification of the self-attention dynamics proposed by Geshkovski et al. (arXiv:2312.10794) to better reflect the practically relevant, causally masked attention used in transformer architectures for generative AI. This modification translates into an interacting particle system that cannot be interpreted as a mean-field gradient flow. Despite this loss of structure, we significantly strengthen the results of Geshkovski et al. (arXiv:2312.10794) in this context: While previous rigorous results focused on cases where all three matrices (Key, Query, and Value) were scaled identities, we prove asymptotic convergence to a single cluster for arbitrary key-query matrices and a value matrix equal to the identity. Additionally, we establish a connection to the classical R'enyi parking problem from combinatorial geometry to make initial theoretical steps towards demonstrating the existence of meta-stable states.