# On the Geometric Structure of Layer Updates in Deep Language Models

arXiv: https://arxiv.org/abs/2604.02459 · HTML: https://mlsanity.com/paper/2604.02459/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 9 references that are in the map.
- First posted on arXiv 2 Apr 2026.

## Builds on (most cited first, 6 of 6)

- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [Probing Classifiers: Promises, Shortcomings, and Advances](https://mlsanity.com/paper/2102.12452/) (2021)
- [The Mythos of Model Interpretability](https://mlsanity.com/paper/1606.03490/) (2016)
- [Probing the Probing Paradigm: Does Probing Accuracy Entail Task Relevance?](https://mlsanity.com/paper/2005.00719/) (2020)
- [DirectProbe: Studying Representations without Classifiers](https://mlsanity.com/paper/2104.05904/) (2021)

## Built on by (most cited first, 2 of 2)

- [Spectral phase transitions and trainability in neural network learning dynamics](https://mlsanity.com/paper/2606.28486/) (2026)
- [Global Divergence, Local Convergence: Representation Geometry in SSMs and Transformers](https://mlsanity.com/paper/2609.08692/) (2026)

## Abstract (arXiv)

We study the geometric structure of layer updates in deep language models. Rather than analyzing what information is encoded in intermediate representations, we ask how representations change from one layer to the next. We show that layerwise updates admit a decomposition into a dominant tokenwise component and a residual that is not captured by restricted tokenwise function classes. Across multiple architectures, including Transformers and state-space models, we find that the full layer update is almost perfectly aligned with the tokenwise component, while the residual exhibits substantially weaker alignment, larger angular deviation, and significantly lower projection onto the dominant tokenwise subspace. This indicates that the residual is not merely a small correction, but a geometrically distinct component of the transformation. This geometric separation has functional consequences: approximation error under the restricted tokenwise model is strongly associated with output perturbation, with Spearman correlations often exceeding 0.7 and reaching up to 0.95 in larger models. Together, these results suggest that most layerwise updates behave like structured reparameterizations along a dominant direction, while functionally significant computation is concentrated in a geometrically distinct residual component. Our framework provides a simple, architecture-agnostic method for probing the geometric and functional structure of layer updates in modern language models.