# Unveiling the Hidden Structure of Self-Attention via Kernel Principal Component Analysis

arXiv: https://arxiv.org/abs/2406.13762 · HTML: https://mlsanity.com/paper/2406.13762/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 71 of its 101 references that are in the map.
- First posted on arXiv 19 Jun 2024.

## Builds on (most cited first, 30 of 71)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [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)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [Training data-efficient image transformers & distillation through attention](https://mlsanity.com/paper/2012.12877/) (2020)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Large Language Models are Zero-Shot Reasoners](https://mlsanity.com/paper/2205.11916/) (2022)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [Towards Evaluating the Robustness of Neural Networks](https://mlsanity.com/paper/1608.04644/) (2016)
- [Benchmarking Neural Network Robustness to Common Corruptions and Perturbations](https://mlsanity.com/paper/1903.12261/) (2019)
- [XLNet: Generalized Autoregressive Pretraining for Language Understanding](https://mlsanity.com/paper/1906.08237/) (2019)
- [An Image is Worth One Word: Personalizing Text-to-Image Generation using Textual Inversion](https://mlsanity.com/paper/2208.01618/) (2022)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [The Many Faces of Robustness: A Critical Analysis of Out-of-Distribution Generalization](https://mlsanity.com/paper/2006.16241/) (2020)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [Decision Transformer: Reinforcement Learning via Sequence Modeling](https://mlsanity.com/paper/2106.01345/) (2021)
- [Linformer: Self-Attention with Linear Complexity](https://mlsanity.com/paper/2006.04768/) (2020)
- [Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks](https://mlsanity.com/paper/2003.01690/) (2020)
- [Natural Adversarial Examples](https://mlsanity.com/paper/1907.07174/) (2019)
- [BERT Rediscovers the Classical NLP Pipeline](https://mlsanity.com/paper/1905.05950/) (2019)
- [Transformers in Vision: A Survey](https://mlsanity.com/paper/2101.01169/) (2021)
- [Analyzing Multi-Head Self-Attention: Specialized Heads Do the Heavy Lifting, the Rest Can Be Pruned](https://mlsanity.com/paper/1905.09418/) (2019)
- [Segmenter: Transformer for Semantic Segmentation](https://mlsanity.com/paper/2105.05633/) (2021)
- [Square Attack: a query-efficient black-box adversarial attack via random search](https://mlsanity.com/paper/1912.00049/) (2019)

## Built on by (most cited first, 7 of 7)

- [Tight Clusters Make Specialized Experts](https://mlsanity.com/paper/2502.15315/) (2025)
- [Graph-Aware Isomorphic Attention for Adaptive Dynamics in Transformers](https://mlsanity.com/paper/2501.02393/) (2025)
- [Transformer Meets Twicing: Harnessing Unattended Residual Information](https://mlsanity.com/paper/2503.00687/) (2025)
- [Rethinking PCA Through Duality](https://mlsanity.com/paper/2510.18130/) (2025)
- [LUNA: Linear Universal Neural Attention with Generalization Guarantees](https://mlsanity.com/paper/2512.08061/) (2025)
- [Self-Attention And Beyond the Infinite: Towards Linear Transformers with Infinite Self-Attention](https://mlsanity.com/paper/2603.00175/) (2026)
- [Anti Mode-Collapse in Mean-Field Transformer via Auxiliary Variables](https://mlsanity.com/paper/2605.30229/) (2026)

## Abstract (arXiv)

The remarkable success of transformers in sequence modeling tasks, spanning various applications in natural language processing and computer vision, is attributed to the critical role of self-attention. Similar to the development of most deep learning models, the construction of these attention mechanisms relies on heuristics and experience. In our work, we derive self-attention from kernel principal component analysis (kernel PCA) and show that self-attention projects its query vectors onto the principal component axes of its key matrix in a feature space. We then formulate the exact formula for the value matrix in self-attention, theoretically and empirically demonstrating that this value matrix captures the eigenvectors of the Gram matrix of the key vectors in self-attention. Leveraging our kernel PCA framework, we propose Attention with Robust Principal Components (RPC-Attention), a novel class of robust attention that is resilient to data contamination. We empirically demonstrate the advantages of RPC-Attention over softmax attention on the ImageNet-1K object classification, WikiText-103 language modeling, and ADE20K image segmentation task.