# Mechanistic Permutability: Match Features Across Layers

arXiv: https://arxiv.org/abs/2410.07656 · HTML: https://mlsanity.com/paper/2410.07656/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Placed in Sparse autoencoder training and evaluation, one of 13 directions in Mechanistic interpretability.
- Builds on 7 of its 11 references that are in the map.
- First posted on arXiv 10 Oct 2024.

## Placed in

- Mechanistic interpretability / [Sparse autoencoder training and evaluation](https://mlsanity.com/mechinterp/sparse-autoencoder-training-and-evaluation/)

## Builds on (most cited first, 7 of 7)

- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [Git Re-Basin: Merging Models modulo Permutation Symmetries](https://mlsanity.com/paper/2209.04836/) (2022)
- [Gemma Scope: Open Sparse Autoencoders Everywhere All At Once on Gemma 2](https://mlsanity.com/paper/2408.05147/) (2024)
- [Finding Neurons in a Haystack: Case Studies with Sparse Probing](https://mlsanity.com/paper/2305.01610/) (2023)
- [Jumping Ahead: Improving Reconstruction Fidelity with JumpReLU Sparse Autoencoders](https://mlsanity.com/paper/2407.14435/) (2024)
- [Linear Algebraic Structure of Word Senses, with Applications to Polysemy](https://mlsanity.com/paper/1601.03764/) (2016)

## Built on by (most cited first, 22 of 22)

- [I Have Covered All the Bases Here: Interpreting Reasoning Features in Large Language Models via Sparse Autoencoders](https://mlsanity.com/paper/2503.18878/) (2025)
- [Tracking the Feature Dynamics in LLM Training: A Mechanistic Study](https://mlsanity.com/paper/2412.17626/) (2024)
- [Analyze Feature Flow to Enhance Interpretation and Steering in Language Models](https://mlsanity.com/paper/2502.03032/) (2025)
- [Evaluating Adversarial Robustness of Concept Representations in Sparse Autoencoders](https://mlsanity.com/paper/2505.16004/) (2025)
- [From Task-Specific Models to Unified Systems: A Review of Model Merging Approaches](https://mlsanity.com/paper/2503.08998/) (2025)
- [Interpretation Meets Safety: A Survey on Interpretation Methods and Tools for Improving LLM Safety](https://mlsanity.com/paper/2506.05451/) (2025)
- [How a Bilingual LM Becomes Bilingual: Tracing Internal Representations with Sparse Autoencoders](https://mlsanity.com/paper/2503.06394/) (2025)
- [The Linear Centroids Hypothesis: Features as Directions Learned by Local Experts](https://mlsanity.com/paper/2604.11962/) (2026)
- [FaithfulSAE: Towards Capturing Faithful Features with Sparse Autoencoders without External Dataset Dependencies](https://mlsanity.com/paper/2506.17673/) (2025)
- [Cross-Layer Discrete Concept Discovery for Interpreting Language Models](https://mlsanity.com/paper/2506.20040/) (2025)
- [Visual Exploration of Feature Relationships in Sparse Autoencoders with Curated Concepts](https://mlsanity.com/paper/2511.06048/) (2025)
- [Semantic Optimal Transport for Sparse Autoencoder Feature Matching and Circuit Compression](https://mlsanity.com/paper/2605.28567/) (2026)
- [The Birth of Knowledge: Emergent Features across Time, Space, and Scale in Large Language Models](https://mlsanity.com/paper/2505.19440/) (2025)
- [Hedonic Neurons: A Mechanistic Mapping of Latent Coalitions in Transformer MLPs](https://mlsanity.com/paper/2509.23684/) (2025)
- [DePass: Unified Feature Attributing by Simple Decomposed Forward Pass](https://mlsanity.com/paper/2510.18462/) (2025)
- [Unstable Features, Reproducible Subspaces: Understanding Seed Dependence in Sparse Autoencoders](https://mlsanity.com/paper/2606.12138/) (2026)
- [Are Single-Token Sparse Autoencoder Features Causally Necessary? Layer-Depth and SAE-Family Effects](https://mlsanity.com/paper/2607.20596/) (2026)
- [CircuitSteer: Geometrically Aligned Multi-Layer Steering via Sparse Autoencoder Circuits](https://mlsanity.com/paper/2608.05732/) (2026)
- [Kronecker Factorization Improves Efficiency and Interpretability of Sparse Autoencoders](https://mlsanity.com/paper/2505.22255/) (2025)
- [fmxcoders: Factorized Masked Crosscoders for Cross-Layer Feature Discovery](https://mlsanity.com/paper/2605.09438/) (2026)
- [Exemplar Partitioning for Mechanistic Interpretability](https://mlsanity.com/paper/2605.14347/) (2026)
- [Beyond a Bag of Features: Set-Level Instability in Sparse Autoencoders](https://mlsanity.com/paper/2608.11197/) (2026)

## Abstract (arXiv)

Understanding how features evolve across layers in deep neural networks is a fundamental challenge in mechanistic interpretability, particularly due to polysemanticity and feature superposition. While Sparse Autoencoders (SAEs) have been used to extract interpretable features from individual layers, aligning these features across layers has remained an open problem. In this paper, we introduce SAE Match, a novel, data-free method for aligning SAE features across different layers of a neural network. Our approach involves matching features by minimizing the mean squared error between the folded parameters of SAEs, a technique that incorporates activation thresholds into the encoder and decoder weights to account for differences in feature scales. Through extensive experiments on the Gemma 2 language model, we demonstrate that our method effectively captures feature evolution across layers, improving feature matching quality. We also show that features persist over several layers and that our approach can approximate hidden states across layers. Our work advances the understanding of feature dynamics in neural networks and provides a new tool for mechanistic interpretability studies.