# Revisiting Anisotropy in Language Transformers: The Geometry of Learning Dynamics

arXiv: https://arxiv.org/abs/2604.08764 · HTML: https://mlsanity.com/paper/2604.08764/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 46 references that are in the map.
- First posted on arXiv 9 Apr 2026.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [How Contextual are Contextualized Word Representations? Comparing the Geometry of BERT, ELMo, and GPT-2 Embeddings](https://mlsanity.com/paper/1909.00512/) (2019)
- [Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning](https://mlsanity.com/paper/2012.13255/) (2020)
- [SmolLM2: When Smol Goes Big -- Data-Centric Training of a Small Language Model](https://mlsanity.com/paper/2502.02737/) (2025)
- [Representation Degeneration Problem in Training Natural Language Generation Models](https://mlsanity.com/paper/1907.12009/) (2019)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [Gradient Descent Happens in a Tiny Subspace](https://mlsanity.com/paper/1812.04754/) (2018)
- [Visualizing and Measuring the Geometry of BERT](https://mlsanity.com/paper/1906.02715/) (2019)
- [All-but-the-Top: Simple and Effective Postprocessing for Word Representations](https://mlsanity.com/paper/1702.01417/) (2017)
- [CRAFT: Concept Recursive Activation FacTorization for Explainability](https://mlsanity.com/paper/2211.10154/) (2022)
- [All Bark and No Bite: Rogue Dimensions in Transformer Language Models Obscure Representational Quality](https://mlsanity.com/paper/2109.04404/) (2021)
- [A Holistic Approach to Unifying Automatic Concept Extraction and Concept Importance Estimation](https://mlsanity.com/paper/2306.07304/) (2023)
- [What needs to go right for an induction head? A mechanistic study of in-context learning circuits and their formation](https://mlsanity.com/paper/2404.07129/) (2024)
- [Anisotropy Is Inherent to Self-Attention in Transformers](https://mlsanity.com/paper/2401.12143/) (2024)
- [IsoScore: Measuring the Uniformity of Embedding Space Utilization](https://mlsanity.com/paper/2108.07344/) (2021)
- [The Shape of Learning: Anisotropy and Intrinsic Dimensions in Transformer-Based Models](https://mlsanity.com/paper/2311.05928/) (2023)
- [A Concept-Based Explainability Framework for Large Multimodal Models](https://mlsanity.com/paper/2406.08074/) (2024)
- [Rare Tokens Degenerate All Tokens: Improving Neural Text Generation via Adaptive Gradient Gating for Rare Token Embeddings](https://mlsanity.com/paper/2109.03127/) (2021)
- [Learning to Remove: Towards Isotropic Pre-trained BERT Embedding](https://mlsanity.com/paper/2104.05274/) (2021)
- [Norm of Word Embedding Encodes Information Gain](https://mlsanity.com/paper/2212.09663/) (2022)
- [Frequency-based Distortions in Contextualized Word Embeddings](https://mlsanity.com/paper/2104.08465/) (2021)
- [Token embeddings violate the manifold hypothesis](https://mlsanity.com/paper/2504.01002/) (2025)
- [Mitigating Frequency Bias and Anisotropy in Language Model Pre-Training with Syntactic Smoothing](https://mlsanity.com/paper/2410.11462/) (2024)
- [Unraveling the Localized Latents: Learning Stratified Manifold Structures in LLM Embedding Space with Sparse Mixture-of-Experts](https://mlsanity.com/paper/2502.13577/) (2025)
- [Beyond Induction Heads: In-Context Meta Learning Induces Multi-Phase Circuit Emergence](https://mlsanity.com/paper/2505.16694/) (2025)
- [Tending Towards Stability: Convergence Challenges in Small Language Models](https://mlsanity.com/paper/2410.11451/) (2024)
- [Estimating the reach of a manifold via its convexity defect function](https://mlsanity.com/paper/2001.08006/) (2020)
- [Isotropy, Clusters, and Classifiers](https://mlsanity.com/paper/2402.03191/) (2024)
- [SGD as Free Energy Minimization: A Thermodynamic View on Neural Network Training](https://mlsanity.com/paper/2505.23489/) (2025)
- [Probing the topology of the space of tokens with structured prompts](https://mlsanity.com/paper/2503.15421/) (2025)

## Built on by (most cited first, 3 of 3)

- [Observable Patterns Are Not Explanations: A Causal-Geometric Analysis of Latent Reasoning Models](https://mlsanity.com/paper/2606.12689/) (2026)
- [Towards Understanding the Power and Limits of the Muon Optimizer: A River-Valley Perspective](https://mlsanity.com/paper/2606.21514/) (2026)
- [When Do Concepts Become Functionally Sufficient During Language-Model Training?](https://mlsanity.com/paper/2608.15323/) (2026)

## Abstract (arXiv)

Since their introduction, Transformer architectures have dominated Natural Language Processing (NLP). However, recent research has highlighted an inherent anisotropy phenomenon in these models, presenting a significant challenge to their geometric interpretation. Previous theoretical studies on this phenomenon are rarely grounded in the underlying representation geometry. In this paper, we extend them by deriving geometric arguments for how frequency-biased sampling attenuates curvature visibility and why training preferentially amplify tangent directions. Empirically, we then use concept-based mechanistic interpretability during training, rather than only post hoc, to fit activation-derived low-rank tangent proxies and test them against ordinary backpropagated true gradients. Across encoder-style and decoder-style language models, we find that these activation-derived directions capture both unusually large gradient energy and a substantially larger share of gradient anisotropy than matched-rank normal controls, providing strong empirical support for a tangent-aligned account of anisotropy.