# How are linear representations learned? Exact solutions to the dynamics of abstraction

arXiv: https://arxiv.org/abs/2607.08843 · HTML: https://mlsanity.com/paper/2607.08843/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Circuits and mechanistic analysis, one of 13 directions in Mechanistic interpretability.
- Builds on 22 of its 47 references that are in the map.
- First posted on arXiv 9 Jul 2026.

## Placed in

- Mechanistic interpretability / [Circuits and mechanistic analysis](https://mlsanity.com/mechinterp/circuits-and-mechanistic-analysis/)

## Builds on (most cited first, 22 of 22)

- [DINOv3](https://mlsanity.com/paper/2508.10104/) (2025)
- [Inference-Time Intervention: Eliciting Truthful Answers from a Language Model](https://mlsanity.com/paper/2306.03341/) (2023)
- [Steering Language Models With Activation Engineering](https://mlsanity.com/paper/2308.10248/) (2023)
- [Gemma: Open Models Based on Gemini Research and Technology](https://mlsanity.com/paper/2403.08295/) (2024)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Finding Neurons in a Haystack: Case Studies with Sparse Probing](https://mlsanity.com/paper/2305.01610/) (2023)
- [Emergent Linear Representations in World Models of Self-Supervised Sequence Models](https://mlsanity.com/paper/2309.00941/) (2023)
- [A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks](https://mlsanity.com/paper/1810.02281/) (2018)
- [The Geometry of Categorical and Hierarchical Concepts in Large Language Models](https://mlsanity.com/paper/2406.01506/) (2024)
- [An analytic theory of generalization dynamics and transfer learning in deep linear networks](https://mlsanity.com/paper/1809.10374/) (2018)
- [Analogies Explained: Towards Understanding Word Embeddings](https://mlsanity.com/paper/1901.09813/) (2019)
- [From Lazy to Rich: Exact Learning Dynamics in Deep Linear Networks](https://mlsanity.com/paper/2409.14623/) (2024)
- [There Will Be a Scientific Theory of Deep Learning](https://mlsanity.com/paper/2604.21691/) (2026)
- [Compositional generalization through abstract representations in human and artificial neural networks](https://mlsanity.com/paper/2209.07431/) (2022)
- [Leveraging the two timescale regime to demonstrate convergence of neural networks](https://mlsanity.com/paper/2304.09576/) (2023)
- [On the Emergence of Linear Analogies in Word Embeddings](https://mlsanity.com/paper/2505.18651/) (2025)
- [Position: Solve Layerwise Linear Models First to Understand Neural Dynamical Phenomena (Neural Collapse, Emergence, Lazy/Rich Regime, and Grokking)](https://mlsanity.com/paper/2502.21009/) (2025)
- [Task structure and nonlinearity jointly determine learned representational geometry](https://mlsanity.com/paper/2401.13558/) (2024)
- [Ultra-fast feature learning for the training of two-layer neural networks in the two-timescale regime](https://mlsanity.com/paper/2504.18208/) (2025)
- [A mathematical theory for understanding when abstract representations emerge in neural networks](https://mlsanity.com/paper/2510.09816/) (2025)

## Abstract (arXiv)

In artificial and biological neural networks, concepts are often encoded as consistent linear directions in representation space. In deep learning, this idea is known as the linear representation hypothesis and underpins many interpretability and control methods based on linear probes, from concept detection to activation steering. Yet while prior work has studied whether such directions should exist textitafter training, the dynamics of how they emerge textitduring training remain poorly understood. Here, we develop a framework to study the alignment of concept directions during training - a process we call "abstraction". In a minimal linear network setting, we obtain exact solutions for the full trajectory of abstraction. These solutions reveal key analytic principles governing abstraction: (i) data and target geometry jointly determine abstraction at the end-of-learning, (ii) abstraction improves with network depth, and (iii) initialization scale controls the maximum abstraction reached during training. Extending our theory to nonlinear networks, we analyze how the choice of nonlinearity affects abstraction dynamics: erf networks approximate the linear theory, while abstraction in ReLU networks depends less on target geometry and more on input geometry. Across both, we prove a striking attenuation law: both nonlinearities weaken abstraction in activations relative to preactivations. We find evidence for this law in open models (DINOv3, Gemma 4) and apply our theory to improve linear probe generalization in LLMs. Together, our results provide a dynamical theory of abstraction with implications for interpretability and control.