# Dimensional Collapse in Transformer Attention Outputs: A Challenge for Sparse Dictionary Learning

arXiv: https://arxiv.org/abs/2508.16929 · HTML: https://mlsanity.com/paper/2508.16929/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Placed in Sparse autoencoder training and evaluation, one of 13 directions in Mechanistic interpretability.
- Builds on 31 of its 52 references that are in the map.
- First posted on arXiv 23 Aug 2025.

## Placed in

- Mechanistic interpretability / [Sparse autoencoder training and evaluation](https://mlsanity.com/mechinterp/sparse-autoencoder-training-and-evaluation/)

## Builds on (most cited first, 30 of 31)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 small](https://mlsanity.com/paper/2211.00593/) (2022)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [Scaling and evaluating sparse autoencoders](https://mlsanity.com/paper/2406.04093/) (2024)
- [Gemma Scope: Open Sparse Autoencoders Everywhere All At Once on Gemma 2](https://mlsanity.com/paper/2408.05147/) (2024)
- [Understanding Dimensional Collapse in Contrastive Self-supervised Learning](https://mlsanity.com/paper/2110.09348/) (2021)
- [Jumping Ahead: Improving Reconstruction Fidelity with JumpReLU Sparse Autoencoders](https://mlsanity.com/paper/2407.14435/) (2024)
- [Understanding self-supervised Learning Dynamics without Contrastive Pairs](https://mlsanity.com/paper/2102.06810/) (2021)
- [A Study of BFLOAT16 for Deep Learning Training](https://mlsanity.com/paper/1905.12322/) (2019)
- [Transcoders Find Interpretable LLM Feature Circuits](https://mlsanity.com/paper/2406.11944/) (2024)
- [Improving Dictionary Learning with Gated Sparse Autoencoders](https://mlsanity.com/paper/2404.16014/) (2024)
- [Linear Algebraic Structure of Word Senses, with Applications to Polysemy](https://mlsanity.com/paper/1601.03764/) (2016)
- [On Feature Decorrelation in Self-Supervised Learning](https://mlsanity.com/paper/2105.00470/) (2021)
- [A Toy Model of Universality: Reverse Engineering How Networks Learn Group Operations](https://mlsanity.com/paper/2302.03025/) (2023)
- [Llama Scope: Extracting Millions of Features from Llama-3.1-8B with Sparse Autoencoders](https://mlsanity.com/paper/2410.20526/) (2024)
- [Universal Neurons in GPT2 Language Models](https://mlsanity.com/paper/2401.12181/) (2024)
- [Low-Rank Bottleneck in Multi-head Attention Models](https://mlsanity.com/paper/2002.07028/) (2020)
- [Successor Heads: Recurring, Interpretable Attention Heads In The Wild](https://mlsanity.com/paper/2312.09230/) (2023)
- [Fixed Encoder Self-Attention Patterns in Transformer-Based Machine Translation](https://mlsanity.com/paper/2002.10260/) (2020)
- [Efficient Dictionary Learning with Switch Sparse Autoencoders](https://mlsanity.com/paper/2410.08201/) (2024)
- [Towards Universality: Studying Mechanistic Similarity Across Language Model Architectures](https://mlsanity.com/paper/2410.06672/) (2024)
- [An X-Ray Is Worth 15 Features: Sparse Autoencoders for Interpretable Radiology Report Generation](https://mlsanity.com/paper/2410.03334/) (2024)
- [Automatically Identifying Local and Global Circuits with Linear Computation Graphs](https://mlsanity.com/paper/2405.13868/) (2024)
- [Small Singular Values Matter: A Random Matrix Analysis of Transformer Models](https://mlsanity.com/paper/2410.17770/) (2024)
- [Sparse Distributed Memory is a Continual Learner](https://mlsanity.com/paper/2303.11934/) (2023)
- [Towards Understanding the Nature of Attention with Low-Rank Sparse Decomposition](https://mlsanity.com/paper/2504.20938/) (2025)

## Built on by (most cited first, 6 of 6)

- [Algebraic Dead Directions in LayerNorm Transformers: A Forward-Pass-Only Diagnostic at LLM Scale](https://mlsanity.com/paper/2606.19491/) (2026)
- [On the Relationship Between Activation Outliers and Feature Death in Sparse Autoencoders](https://mlsanity.com/paper/2605.31518/) (2026)
- [When and How Long? The Readout-Mediator Angle in Temporal Reasoning](https://mlsanity.com/paper/2605.29126/) (2026)
- [Unstable Features, Reproducible Subspaces: Understanding Seed Dependence in Sparse Autoencoders](https://mlsanity.com/paper/2606.12138/) (2026)
- [Radial Suppression Accelerates Algorithmic Generalization: A Geometric Analysis of Delayed Generalization](https://mlsanity.com/paper/2606.32000/) (2026)
- [Size Doesn't Matter: Cosine-Scored Sparse Autoencoders](https://mlsanity.com/paper/2606.15054/) (2026)

## Abstract (arXiv)

Transformer architectures, and their attention mechanisms in particular, form the foundation of modern large language models. While transformer models are widely believed to operate in high-dimensional hidden spaces, we show that attention outputs are in fact confined to a surprisingly low-dimensional subspace, with an effective dimensionality of only about 60% of the full space. In contrast, MLP outputs and residual streams remain much closer to full-rank, exhibiting effective ranks around 90%. This striking dimensional discrepancy is consistently observed across diverse model families and datasets, and is strongly shaped by the attention output projection matrix. Critically, we find this low-rank structure as a key factor of the prevalent dead feature problem in sparse dictionary learning, where it creates a mismatch between randomly initialized features and the intrinsic geometry of the activation space. Building on this insight, we propose a subspace-constrained training method for sparse autoencoders (SAEs), initializing feature directions into the active subspace of activations. Our approach reduces dead features from 87% to below 1% in Attention Output SAEs with 1M features, and can further extend to other sparse dictionary learning methods. Our findings provide both new insights into the geometry of attention and practical tools for improving sparse dictionary learning in large language models.