# Sparsification and Reconstruction from the Perspective of Representation Geometry

arXiv: https://arxiv.org/abs/2505.22506 · HTML: https://mlsanity.com/paper/2505.22506/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Sparse autoencoder training and evaluation, one of 13 directions in Mechanistic interpretability.
- Builds on 34 of its 46 references that are in the map.
- First posted on arXiv 28 May 2025.

## Placed in

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

## Builds on (most cited first, 30 of 34)

- [Gemma 2: Improving Open Language Models at a Practical Size](https://mlsanity.com/paper/2408.00118/) (2024)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [How Contextual are Contextualized Word Representations? Comparing the Geometry of BERT, ELMo, and GPT-2 Embeddings](https://mlsanity.com/paper/1909.00512/) (2019)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [Towards Automated Circuit Discovery for Mechanistic Interpretability](https://mlsanity.com/paper/2304.14997/) (2023)
- [Mechanistic Interpretability for AI Safety -- A Review](https://mlsanity.com/paper/2404.14082/) (2024)
- [Not All Language Model Features Are One-Dimensionally Linear](https://mlsanity.com/paper/2405.14860/) (2024)
- [Open Problems in Mechanistic Interpretability](https://mlsanity.com/paper/2501.16496/) (2025)
- [Improving Dictionary Learning with Gated Sparse Autoencoders](https://mlsanity.com/paper/2404.16014/) (2024)
- [BatchTopK Sparse Autoencoders](https://mlsanity.com/paper/2412.06410/) (2024)
- [Disentangled Representation Learning](https://mlsanity.com/paper/2211.11695/) (2022)
- [The Geometry of Multilingual Language Model Representations](https://mlsanity.com/paper/2205.10964/) (2022)
- [On the Origins of Linear Representations in Large Language Models](https://mlsanity.com/paper/2403.03867/) (2024)
- [Transformers represent belief state geometry in their residual stream](https://mlsanity.com/paper/2405.15943/) (2024)
- [A Survey on Sparse Autoencoders: Interpreting the Internal Mechanisms of Large Language Models](https://mlsanity.com/paper/2503.05613/) (2025)
- [Steering Knowledge Selection Behaviours in LLMs via SAE-Based Representation Engineering](https://mlsanity.com/paper/2410.15999/) (2024)
- [Guided Variational Autoencoder for Disentanglement Learning](https://mlsanity.com/paper/2004.01255/) (2020)
- [The Geometry of Concepts: Sparse Autoencoder Feature Structure](https://mlsanity.com/paper/2410.19750/) (2024)
- [Multi-VAE: Learning Disentangled View-common and View-peculiar Visual Representations for Multi-view Clustering](https://mlsanity.com/paper/2106.11232/) (2021)
- [Measuring Progress in Dictionary Learning for Language Model Interpretability with Board Game Models](https://mlsanity.com/paper/2408.00113/) (2024)
- [Interpreting Attention Layer Outputs with Sparse Autoencoders](https://mlsanity.com/paper/2406.17759/) (2024)
- [Efficient Dictionary Learning with Switch Sparse Autoencoders](https://mlsanity.com/paper/2410.08201/) (2024)
- [Diagonal Attention and Style-based GAN for Content-Style Disentanglement in Image Generation and Translation](https://mlsanity.com/paper/2103.16146/) (2021)
- [Superposition Yields Robust Neural Scaling](https://mlsanity.com/paper/2505.10465/) (2025)
- [OOGAN: Disentangling GAN with One-Hot Sampling and Orthogonal Regularization](https://mlsanity.com/paper/1905.10836/) (2019)
- [JumpReLU: A Retrofit Defense Strategy for Adversarial Attacks](https://mlsanity.com/paper/1904.03750/) (2019)
- [Group-SAE: Efficient Training of Sparse Autoencoders for Large Language Models via Layer Groups](https://mlsanity.com/paper/2410.21508/) (2024)
- [Fast Parallel Algorithms for Euclidean Minimum Spanning Tree and Hierarchical Spatial Clustering](https://mlsanity.com/paper/2104.01126/) (2021)
- [Adaptive Sparse Allocation with Mutual Choice & Feature Choice Sparse Autoencoders](https://mlsanity.com/paper/2411.02124/) (2024)

## Abstract (arXiv)

Sparse Autoencoders (SAEs) have emerged as a predominant tool in mechanistic interpretability, aiming to identify interpretable monosemantic features. However, how does sparse encoding organize the representations of activation vector from language models? What is the relationship between this organizational paradigm and feature disentanglement as well as reconstruction performance? To address these questions, we propose the SAEMA, which validates the stratified structure of the representation by observing the variability of the rank of the symmetric semipositive definite (SSPD) matrix corresponding to the modal tensor unfolded along the latent tensor with the level of noise added to the residual stream. To systematically investigate how sparse encoding alters representational structures, we define local and global representations, demonstrating that they amplify inter-feature distinctions by merging similar semantic features and introducing additional dimensionality. Furthermore, we intervene the global representation from an optimization perspective, proving a significant causal relationship between their separability and the reconstruction performance. This study explains the principles of sparsity from the perspective of representational geometry and demonstrates the impact of changes in representational structure on reconstruction performance. Particularly emphasizes the necessity of understanding representations and incorporating representational constraints, providing empirical references for developing new interpretable tools and improving SAEs. The code is available at hyperlinkhttps://github.com/wenjie1835/SAERepGeohttps://github.com/wenjie1835/SAERepGeo.