# Tracing the Representation Geometry of Language Models from Pretraining to Post-training

arXiv: https://arxiv.org/abs/2509.23024 · HTML: https://mlsanity.com/paper/2509.23024/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 54 references that are in the map.
- First posted on arXiv 27 Sep 2025.

## Builds on (most cited first, 30 of 45)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [TriviaQA: A Large Scale Distantly Supervised Challenge Dataset for Reading Comprehension](https://mlsanity.com/paper/1705.03551/) (2017)
- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [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)
- [The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale](https://mlsanity.com/paper/2406.17557/) (2024)
- [Tulu 3: Pushing Frontiers in Open Language Model Post-Training](https://mlsanity.com/paper/2411.15124/) (2024)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [Crowdsourcing Multiple Choice Science Questions](https://mlsanity.com/paper/1707.06209/) (2017)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [OLMo: Accelerating the Science of Language Models](https://mlsanity.com/paper/2402.00838/) (2024)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [2 OLMo 2 Furious](https://mlsanity.com/paper/2501.00656/) (2024)
- [Gradient Starvation: A Learning Proclivity in Neural Networks](https://mlsanity.com/paper/2011.09468/) (2020)
- [Predictability and Surprise in Large Generative Models](https://mlsanity.com/paper/2202.07785/) (2022)
- [SPoC: Search-based Pseudocode to Code](https://mlsanity.com/paper/1906.04908/) (2019)
- [RankMe: Assessing the downstream performance of pretrained self-supervised representations by their rank](https://mlsanity.com/paper/2210.02885/) (2022)
- [Infini-gram: Scaling Unbounded n-gram Language Models to a Trillion Tokens](https://mlsanity.com/paper/2401.17377/) (2024)
- [The Quantization Model of Neural Scaling](https://mlsanity.com/paper/2303.13506/) (2023)
- [Echo Chamber: RL Post-training Amplifies Behaviors Learned in Pretraining](https://mlsanity.com/paper/2504.07912/) (2025)
- [Learning Dynamics of LLM Finetuning](https://mlsanity.com/paper/2407.10490/) (2024)

## Built on by (most cited first, 30 of 35)

- [The Geometry of Reasoning: Flowing Logics in Representation Space](https://mlsanity.com/paper/2510.09782/) (2025)
- [Context Structure Reshapes the Representational Geometry of Language Models](https://mlsanity.com/paper/2601.22364/) (2026)
- [Towards a Mechanistic Understanding of Large Reasoning Models: A Survey of Training, Inference, and Failures](https://mlsanity.com/paper/2601.19928/) (2026)
- [Spectral Lens: Activation and Gradient Spectra as Diagnostics of LLM Optimization](https://mlsanity.com/paper/2605.05683/) (2026)
- [Convergence and Divergence of Language Models under Different Random Seeds](https://mlsanity.com/paper/2509.26643/) (2025)
- [Student Guides Teacher: Weak-to-Strong Inference via Spectral Orthogonal Exploration](https://mlsanity.com/paper/2601.06160/) (2026)
- [Disentangling Geometry, Performance, and Training in Language Models](https://mlsanity.com/paper/2602.20433/) (2026)
- [Uncovering the Latent Potential of Deep Intermediate Representations](https://mlsanity.com/paper/2605.23033/) (2026)
- [Scalable Visual Pretraining for Language Intelligence](https://mlsanity.com/paper/2607.09657/) (2026)
- [Geometric Stability: The Missing Axis of Representations](https://mlsanity.com/paper/2601.09173/) (2026)
- [Dispersion Loss Counteracts Embedding Condensation and Improves Generalization in Small Language Models](https://mlsanity.com/paper/2602.00217/) (2026)
- [Convergent World Representations and Divergent Tasks](https://mlsanity.com/paper/2602.00533/) (2026)
- [From Dispersion to Attraction: Spectral Dynamics of Hallucination Across Whisper Model Scales](https://mlsanity.com/paper/2604.08591/) (2026)
- [Scale Determines Whether Language Models Organize Representation Geometry for Prediction](https://mlsanity.com/paper/2605.17084/) (2026)
- [Structure Before Collapse: Transient semantic geometry in next-token prediction](https://mlsanity.com/paper/2606.26749/) (2026)
- [Logic Before Language: Pre-pretraining on Formal Derivations Fosters Skill Acquisition and Compressibility](https://mlsanity.com/paper/2608.03930/) (2026)
- [The Geometric Anatomy of Capability Acquisition in Transformers](https://mlsanity.com/paper/2602.15997/) (2026)
- [Progressive Residual Warmup for Language Model Pretraining](https://mlsanity.com/paper/2603.05369/) (2026)
- [On the Blessing of Pre-training in Weak-to-Strong Generalization](https://mlsanity.com/paper/2605.05710/) (2026)
- [Layer-wise Representation Dynamics: An Empirical Investigation Across Embedders and Base LLMs](https://mlsanity.com/paper/2605.12714/) (2026)
- [Uncovering the Representation Geometry of Minimal Cores in Overcomplete Reasoning Traces](https://mlsanity.com/paper/2605.14358/) (2026)
- [A Pre-Training Analogue of Grokking in Language Models: Tracing Delayed Grammatical Generalization](https://mlsanity.com/paper/2606.00230/) (2026)
- [Rewriting or Reweighting? A Geometric Account in Language Models](https://mlsanity.com/paper/2608.01835/) (2026)
- [Erased, Rerouted, or Rescaled? Post-Training and the Causal Quotient of a Language Model's Belief State](https://mlsanity.com/paper/2610.05292/) (2026)
- [False Fixed Points: Kantian Feedback, Stable Miscalibration, and Representational Compression in LLMs](https://mlsanity.com/paper/2510.14925/) (2025)
- [Balancing Saliency and Coverage: Semantic Prominence-Aware Budgeting for Visual Token Compression in VLMs](https://mlsanity.com/paper/2603.14892/) (2026)
- [GeoLAN: Geometric Learning of Latent Explanatory Directions in Large Language Models](https://mlsanity.com/paper/2603.19460/) (2026)
- [LLM Pretraining Shapes a Generalizable Manifold: Insights into Cross-Modal Transfer to Time Series](https://mlsanity.com/paper/2605.20449/) (2026)
- [LaRA: Layer-wise Representation Analysis for Detecting Data Contamination in RL Post-Training](https://mlsanity.com/paper/2605.29888/) (2026)
- [Neural Collapse Is Forbidden: Information Floors in Language Models](https://mlsanity.com/paper/2607.09487/) (2026)

## Abstract (arXiv)

Standard training metrics like loss fail to explain the emergence of complex capabilities in large language models. We take a spectral approach to investigate the geometry of learned representations across pretraining and post-training, measuring effective rank (RankMe) and eigenspectrum decay (α-ReQ). With OLMo (1B-7B) and Pythia (160M-12B) models, we uncover a consistent non-monotonic sequence of three geometric phases during autoregressive pretraining. The initial "warmup" phase exhibits rapid representational collapse. This is followed by an "entropy-seeking" phase, where the manifold's dimensionality expands substantially, coinciding with peak n-gram memorization. Subsequently, a "compression-seeking" phase imposes anisotropic consolidation, selectively preserving variance along dominant eigendirections while contracting others, a transition marked with significant improvement in downstream task performance. We show these phases can emerge from a fundamental interplay of cross-entropy optimization under skewed token frequencies and representational bottlenecks (d ll |V|). Post-training further transforms geometry: SFT and DPO drive "entropy-seeking" dynamics to integrate specific instructional or preferential data, improving in-distribution performance while degrading out-of-distribution robustness. Conversely, RLVR induces "compression-seeking", enhancing reward alignment but reducing generation diversity.