# Hidden Holes: topological aspects of language models

arXiv: https://arxiv.org/abs/2406.05798 · HTML: https://mlsanity.com/paper/2406.05798/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 24 references that are in the map.
- First posted on arXiv 9 Jun 2024.

## Builds on (most cited first, 10 of 10)

- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [Deduplicating Training Data Makes Language Models Better](https://mlsanity.com/paper/2107.06499/) (2021)
- [Language Modeling with Gated Convolutional Networks](https://mlsanity.com/paper/1612.08083/) (2016)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Regularizing and Optimizing LSTM Language Models](https://mlsanity.com/paper/1708.02182/) (2017)
- [Scaling Transformer to 1M tokens and beyond with RMT](https://mlsanity.com/paper/2304.11062/) (2023)

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

- [Topological Data Analysis Applications in Natural Language Processing: A Survey](https://mlsanity.com/paper/2411.10298/) (2024)
- [Trust in One Round: Confidence Estimation for Large Language Models via Structural Signals](https://mlsanity.com/paper/2602.00977/) (2026)
- [When Annotators Disagree, Topology Explains: Mapper, a Topological Tool for Exploring Text Embedding Geometry and Ambiguity](https://mlsanity.com/paper/2510.17548/) (2025)
- [Testing the Machine Consciousness Hypothesis](https://mlsanity.com/paper/2512.01081/) (2025)
- [The Shape of Addition: Geometric Structures of Arithmetic in Large Language Models](https://mlsanity.com/paper/2606.03645/) (2026)
- [Topological Attribution Distance (TAD): Revealing Segment-Level RAG Influence on LLM Output Geometry for Incident Log Analysis](https://mlsanity.com/paper/2608.16775/) (2026)

## Abstract (arXiv)

We explore the topology of representation manifolds arising in autoregressive neural language models trained on raw text data. In order to study their properties, we introduce tools from computational algebraic topology, which we use as a basis for a measure of topological complexity, that we call perforation. Using this measure, we study the evolution of topological structure in GPT based large language models across depth and time during training. We then compare these to gated recurrent models, and show that the latter exhibit more topological complexity, with a distinct pattern of changes common to all natural languages but absent from synthetically generated data. The paper presents a detailed analysis of the representation manifolds derived by these models based on studying the shapes of vector clouds induced by them as they are conditioned on sentences from corpora of natural language text. The methods developed in this paper are novel in the field and based on mathematical apparatus that might be unfamiliar to the target audience. To help with that we introduce the minimum necessary theory, and provide additional visualizations in the appendices. The main contribution of the paper is a striking observation about the topological structure of the transformer as compared to LSTM based neural architectures. It suggests that further research into mathematical properties of these neural networks is necessary to understand the operation of large transformer language models. We hope this work inspires further explorations in this direction within the NLP community.