# Persistent Topological Features in Large Language Models

arXiv: https://arxiv.org/abs/2410.11042 · HTML: https://mlsanity.com/paper/2410.11042/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 82 references that are in the map.
- First posted on arXiv 14 Oct 2024.

## Builds on (most cited first, 30 of 50)

- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [The Pile: An 800GB Dataset of Diverse Text for Language Modeling](https://mlsanity.com/paper/2101.00027/) (2020)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [LLM-Pruner: On the Structural Pruning of Large Language Models](https://mlsanity.com/paper/2305.11627/) (2023)
- [Reducing Transformer Depth on Demand with Structured Dropout](https://mlsanity.com/paper/1909.11556/) (2019)
- [ShortGPT: Layers in Large Language Models are More Redundant Than You Expect](https://mlsanity.com/paper/2403.03853/) (2024)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019)
- [The Unreasonable Ineffectiveness of the Deeper Layers](https://mlsanity.com/paper/2403.17887/) (2024)
- [Not All Language Model Features Are One-Dimensionally Linear](https://mlsanity.com/paper/2405.14860/) (2024)
- [From Words to Watts: Benchmarking the Energy Costs of Large Language Model Inference](https://mlsanity.com/paper/2310.03003/) (2023)
- [The Remarkable Robustness of LLMs: Stages of Inference?](https://mlsanity.com/paper/2406.19384/) (2024)
- [Similarity of Neural Network Models: A Survey of Functional and Representational Measures](https://mlsanity.com/paper/2305.06329/) (2023)
- [AI Transparency in the Age of LLMs: A Human-Centered Research Roadmap](https://mlsanity.com/paper/2306.01941/) (2023)
- [On the Effect of Dropping Layers of Pre-trained Transformer Models](https://mlsanity.com/paper/2004.03844/) (2020)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [The geometry of hidden representations of large transformer models](https://mlsanity.com/paper/2302.00294/) (2023)
- [Intrinsic Dimension Estimation for Robust Detection of AI-Generated Texts](https://mlsanity.com/paper/2306.04723/) (2023)
- [Accelerating Training of Transformer-Based Language Models with Progressive Layer Dropping](https://mlsanity.com/paper/2010.13369/) (2020)
- [What Matters in Transformers? Not All Attention is Needed](https://mlsanity.com/paper/2406.15786/) (2024)
- [Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology](https://mlsanity.com/paper/1812.09764/) (2018)
- [Beyond Efficiency: A Systematic Survey of Resource-Efficient Large Language Models](https://mlsanity.com/paper/2401.00625/) (2024)
- [Representation Topology Divergence: A Method for Comparing Neural Network Representations](https://mlsanity.com/paper/2201.00058/) (2021)
- [Emergence of a High-Dimensional Abstraction Phase in Language Transformers](https://mlsanity.com/paper/2405.15471/) (2024)
- [Stochastic Convergence of Persistence Landscapes and Silhouettes](https://mlsanity.com/paper/1312.0308/) (2013)

## Built on by (most cited first, 12 of 12)

- [The Intrinsic Dimension of Prompts in Internal Representations of Large Language Models](https://mlsanity.com/paper/2501.10573/) (2025)
- [The Shape of Adversarial Influence: Characterizing LLM Latent Spaces with Persistent Homology](https://mlsanity.com/paper/2505.20435/) (2025)
- [From Topology to Retrieval: Decoding Embedding Spaces with Unified Signatures](https://mlsanity.com/paper/2511.22150/) (2025)
- [Large Vision-Language Models Get Lost in Attention](https://mlsanity.com/paper/2605.05668/) (2026)
- [Topological Data Analysis Applications in Natural Language Processing: A Survey](https://mlsanity.com/paper/2411.10298/) (2024)
- [CURing Large Models: Compression via CUR Decomposition](https://mlsanity.com/paper/2501.04211/) (2025)
- [From Persistence to Survival: Hypothesis Testing, Effect Sizes and Vectorisation for Topological Features](https://mlsanity.com/paper/2606.11911/) (2026)
- [Optimizing Soft Prompt Tuning via Structural Evolution](https://mlsanity.com/paper/2602.16500/) (2026)
- [TIPS: Topological Ill-Posedness Probing and Steering in Large Language Models](https://mlsanity.com/paper/2606.23590/) (2026)
- [Topological Signatures of Context-Level Reliability in TabPFN](https://mlsanity.com/paper/2607.17962/) (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)
- [The sublevel Flood bifiltration: towards scalable 2-parameter persistent homology](https://mlsanity.com/paper/2610.05441/) (2026)

## Abstract (arXiv)

Understanding the decision-making processes of large language models is critical given their widespread applications. To achieve this, we aim to connect a formal mathematical framework - zigzag persistence from topological data analysis - with practical and easily applicable algorithms. Zigzag persistence is particularly effective for characterizing data as it dynamically transforms across model layers. Within this framework, we introduce topological descriptors that measure how topological features, p-dimensional holes, persist and evolve throughout the layers. Unlike methods that assess each layer individually and then aggregate the results, our approach directly tracks the full evolutionary path of these features. This offers a statistical perspective on how prompts are rearranged and their relative positions changed in the representation space, providing insights into the system's operation as an integrated whole. To demonstrate the expressivity and applicability of our framework, we highlight how sensitive these descriptors are to different models and a variety of datasets. As a showcase application to a downstream task, we use zigzag persistence to establish a criterion for layer pruning, achieving results comparable to state-of-the-art methods while preserving the system-level perspective.