# Spectral Principal Paths: A Spectral Perspective on Linear Representation Formation in LLMs

arXiv: https://arxiv.org/abs/2506.08543 · HTML: https://mlsanity.com/paper/2506.08543/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 35 of its 42 references that are in the map.
- First posted on arXiv 10 Jun 2025.

## Builds on (most cited first, 30 of 35)

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- [DINOv2: Learning Robust Visual Features without Supervision](https://mlsanity.com/paper/2304.07193/) (2023)
- [Emerging Properties in Self-Supervised Vision Transformers](https://mlsanity.com/paper/2104.14294/) (2021)
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- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [TruthfulQA: Measuring How Models Mimic Human Falsehoods](https://mlsanity.com/paper/2109.07958/) (2021)
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- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
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- [SimCSE: Simple Contrastive Learning of Sentence Embeddings](https://mlsanity.com/paper/2104.08821/) (2021)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Representation Engineering: A Top-Down Approach to AI Transparency](https://mlsanity.com/paper/2310.01405/) (2023)
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- [Striving for Simplicity: The All Convolutional Net](https://mlsanity.com/paper/1412.6806/) (2014)
- [Ethical and social risks of harm from Language Models](https://mlsanity.com/paper/2112.04359/) (2021)
- [Probing Classifiers: Promises, Shortcomings, and Advances](https://mlsanity.com/paper/2102.12452/) (2021)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Towards Automated Circuit Discovery for Mechanistic Interpretability](https://mlsanity.com/paper/2304.14997/) (2023)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [Scaling Vision Transformers to 22 Billion Parameters](https://mlsanity.com/paper/2302.05442/) (2023)
- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)

## Abstract (arXiv)

High-level representations have become a central focus in enhancing AI transparency and control, shifting attention from individual neurons or circuits to structured semantic directions that align with human-interpretable concepts. While the Linear Representation Hypothesis (LRH) suggests that such directions emerge in representations, it remains unclear how these representations originate and why they become increasingly stable across layers. To solve this issue, we introduce the Input-Space Linearity Hypothesis, positing that concept-aligned directions originate in the input space and are steadily maintained with increasing depth. We then propose the Spectral Principal Path (SPP) framework, which formalizes how deep networks progressively distill linear representations along the spectral principal directions. We provide rigorous stability guarantees for the SPP based on the Wedin sinTheta perturbation theorem, identifying testable conditions, including spectral gap and context incoherence, that jointly ensure layer-wise directional preservation. By bridging theoretical analysis with empirical evidence, this work identifies a spectral view of how linear representations arise in LLMs, and suggests potential implications for concept-level controllable, robust, and coherent approaches to fairness and transparency in modern AI systems.