# Self-attention Networks Localize When QK-eigenspectrum Concentrates

arXiv: https://arxiv.org/abs/2402.02098 · HTML: https://mlsanity.com/paper/2402.02098/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 31 references that are in the map.
- First posted on arXiv 3 Feb 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)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Training data-efficient image transformers & distillation through attention](https://mlsanity.com/paper/2012.12877/) (2020)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [fairseq: A Fast, Extensible Toolkit for Sequence Modeling](https://mlsanity.com/paper/1904.01038/) (2019)
- [An Explanation of In-context Learning as Implicit Bayesian Inference](https://mlsanity.com/paper/2111.02080/) (2021)
- [On Layer Normalization in the Transformer Architecture](https://mlsanity.com/paper/2002.04745/) (2020)
- [Generating Sequences With Recurrent Neural Networks](https://mlsanity.com/paper/1308.0850/) (2013)
- [Attention is Not All You Need: Pure Attention Loses Rank Doubly Exponentially with Depth](https://mlsanity.com/paper/2103.03404/) (2021)
- [From Softmax to Sparsemax: A Sparse Model of Attention and Multi-Label Classification](https://mlsanity.com/paper/1602.02068/) (2016)
- [ReZero is All You Need: Fast Convergence at Large Depth](https://mlsanity.com/paper/2003.04887/) (2020)
- [Stabilizing Transformer Training by Preventing Attention Entropy Collapse](https://mlsanity.com/paper/2303.06296/) (2023)
- [Birth of a Transformer: A Memory Viewpoint](https://mlsanity.com/paper/2306.00802/) (2023)
- [Signal Propagation in Transformers: Theoretical Perspectives and the Role of Rank Collapse](https://mlsanity.com/paper/2206.03126/) (2022)
- [Scan and Snap: Understanding Training Dynamics and Token Composition in 1-layer Transformer](https://mlsanity.com/paper/2305.16380/) (2023)
- [Vision Transformers provably learn spatial structure](https://mlsanity.com/paper/2210.09221/) (2022)
- [Transformers as Support Vector Machines](https://mlsanity.com/paper/2308.16898/) (2023)
- [Max-Margin Token Selection in Attention Mechanism](https://mlsanity.com/paper/2306.13596/) (2023)
- [Grokking of Hierarchical Structure in Vanilla Transformers](https://mlsanity.com/paper/2305.18741/) (2023)
- [JoMA: Demystifying Multilayer Transformers via JOint Dynamics of MLP and Attention](https://mlsanity.com/paper/2310.00535/) (2023)
- [Spike No More: Stabilizing the Pre-training of Large Language Models](https://mlsanity.com/paper/2312.16903/) (2023)
- [Characterizing Intrinsic Compositionality in Transformers with Tree Projections](https://mlsanity.com/paper/2211.01288/) (2022)
- [Analyzing Feed-Forward Blocks in Transformers through the Lens of Attention Maps](https://mlsanity.com/paper/2302.00456/) (2023)
- [What they do when in doubt: a study of inductive biases in seq2seq learners](https://mlsanity.com/paper/2006.14953/) (2020)
- [B2T Connection: Serving Stability and Performance in Deep Transformers](https://mlsanity.com/paper/2206.00330/) (2022)

## Built on by (most cited first, 19 of 19)

- [Dynamic metastability in the self-attention model](https://mlsanity.com/paper/2410.06833/) (2024)
- [Measure-to-measure interpolation using Transformers](https://mlsanity.com/paper/2411.04551/) (2024)
- [Gating is Weighting: Understanding Gated Linear Attention through In-context Learning](https://mlsanity.com/paper/2504.04308/) (2025)
- [Theoretical Insights into Fine-Tuning Attention Mechanism: Generalization and Optimization](https://mlsanity.com/paper/2410.02247/) (2024)
- [Optimizing Attention with Mirror Descent: Generalized Max-Margin Token Selection](https://mlsanity.com/paper/2410.14581/) (2024)
- [Understanding Catastrophic Forgetting In LoRA via Mean-Field Attention Dynamics](https://mlsanity.com/paper/2402.15415/) (2024)
- [AERO: Entropy-Guided Framework for Private LLM Inference](https://mlsanity.com/paper/2410.13060/) (2024)
- [Mitigating Attention Localization in Small Scale: Self-Attention Refinement via One-step Belief Propagation](https://mlsanity.com/paper/2509.07324/) (2025)
- [NerVE: Nonlinear Eigenspectrum Dynamics in LLM Feed-Forward Networks](https://mlsanity.com/paper/2603.06922/) (2026)
- [Gradient Flow Polarizes Softmax Outputs towards Low-Entropy Solutions](https://mlsanity.com/paper/2603.06248/) (2026)
- [Mechanism-Driven Monitors for Preemptive Detection of LLM Training Instability](https://mlsanity.com/paper/2606.28116/) (2026)
- [A Random Matrix Theory Perspective on the Learning Dynamics of Multi-head Latent Attention](https://mlsanity.com/paper/2507.09394/) (2025)
- [On the Role of Hidden States of Modern Hopfield Network in Transformer](https://mlsanity.com/paper/2511.20698/) (2025)
- [Mitigating Object Hallucinations in LVLMs via Attention Imbalance Rectification](https://mlsanity.com/paper/2603.24058/) (2026)
- [QUEST: A robust attention formulation using query-modulated spherical attention](https://mlsanity.com/paper/2604.00199/) (2026)
- [Faster Query-Key Learning Sharpens Attention in Self-Attention Models](https://mlsanity.com/paper/2608.06776/) (2026)
- [Dispelling the Curse of Singularities in Neural Network Optimizations](https://mlsanity.com/paper/2602.01308/) (2026)
- [Anti Mode-Collapse in Mean-Field Transformer via Auxiliary Variables](https://mlsanity.com/paper/2605.30229/) (2026)
- [Query Expansion and Key Specialization in Transformer Attention Geometry](https://mlsanity.com/paper/2609.34273/) (2026)

## Abstract (arXiv)

The self-attention mechanism prevails in modern machine learning. It has an interesting functionality of adaptively selecting tokens from an input sequence by modulating the degree of attention localization, which many researchers speculate is the basis of the powerful model performance but complicates the underlying mechanism of the learning dynamics. In recent years, mainly two arguments have connected attention localization to the model performances. One is the rank collapse, where the embedded tokens by a self-attention block become very similar across different tokens, leading to a less expressive network. The other is the entropy collapse, where the attention probability approaches non-uniform and entails low entropy, making the learning dynamics more likely to be trapped in plateaus. These two failure modes may apparently contradict each other because the rank and entropy collapses are relevant to uniform and non-uniform attention, respectively. To this end, we characterize the notion of attention localization by the eigenspectrum of query-key parameter matrices and reveal that a small eigenspectrum variance leads attention to be localized. Interestingly, the small eigenspectrum variance prevents both rank and entropy collapse, leading to better model expressivity and trainability.