# Understanding sparse autoencoder scaling in the presence of feature manifolds

arXiv: https://arxiv.org/abs/2509.02565 · HTML: https://mlsanity.com/paper/2509.02565/

## Key facts

- Cited by 11 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 21 of its 30 references that are in the map.
- First posted on arXiv 2 Sep 2025.

## Placed in

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

## Builds on (most cited first, 21 of 21)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [Scaling and evaluating sparse autoencoders](https://mlsanity.com/paper/2406.04093/) (2024)
- [Gemma Scope: Open Sparse Autoencoders Everywhere All At Once on Gemma 2](https://mlsanity.com/paper/2408.05147/) (2024)
- [Jumping Ahead: Improving Reconstruction Fidelity with JumpReLU Sparse Autoencoders](https://mlsanity.com/paper/2407.14435/) (2024)
- [Not All Language Model Features Are One-Dimensionally Linear](https://mlsanity.com/paper/2405.14860/) (2024)
- [Transcoders Find Interpretable LLM Feature Circuits](https://mlsanity.com/paper/2406.11944/) (2024)
- [A is for Absorption: Studying Feature Splitting and Absorption in Sparse Autoencoders](https://mlsanity.com/paper/2409.14507/) (2024)
- [The Quantization Model of Neural Scaling](https://mlsanity.com/paper/2303.13506/) (2023)
- [BatchTopK Sparse Autoencoders](https://mlsanity.com/paper/2412.06410/) (2024)
- [Archetypal SAE: Adaptive and Stable Dictionary Learning for Concept Extraction in Large Vision Models](https://mlsanity.com/paper/2502.12892/) (2025)
- [Projecting Assumptions: The Duality Between Sparse Autoencoders and Concept Geometry](https://mlsanity.com/paper/2503.01822/) (2025)
- [The Origins of Representation Manifolds in Large Language Models](https://mlsanity.com/paper/2505.18235/) (2025)
- [From Flat to Hierarchical: Extracting Sparse Representations with Matching Pursuit](https://mlsanity.com/paper/2506.03093/) (2025)
- [Recurrent Neural Networks Learn to Store and Generate Sequences using Non-Linear Representations](https://mlsanity.com/paper/2408.10920/) (2024)
- [Decomposing The Dark Matter of Sparse Autoencoders](https://mlsanity.com/paper/2410.14670/) (2024)
- [Neural Scaling Laws Rooted in the Data Distribution](https://mlsanity.com/paper/2412.07942/) (2024)
- [Incorporating Hierarchical Semantics in Sparse Autoencoder Architectures](https://mlsanity.com/paper/2506.01197/) (2025)
- [Group Crosscoders for Mechanistic Analysis of Symmetry](https://mlsanity.com/paper/2410.24184/) (2024)

## Built on by (most cited first, 11 of 11)

- [When Models Manipulate Manifolds: The Geometry of a Counting Task](https://mlsanity.com/paper/2601.04480/) (2026)
- [Do Sparse Autoencoders Capture Concept Manifolds?](https://mlsanity.com/paper/2604.28119/) (2026)
- [SynthSAEBench: Evaluating Sparse Autoencoders on Scalable Realistic Synthetic Data](https://mlsanity.com/paper/2602.14687/) (2026)
- [Why Larger Models Learn More: Effects of Capacity, Interference, and Rare-Task Retention](https://mlsanity.com/paper/2605.29548/) (2026)
- [Structuring Sparsity: Block-Sparse Featurizers Capture Visual Concept Manifolds](https://mlsanity.com/paper/2606.25234/) (2026)
- [Subspace-Aware Sparse Autoencoders for Effective Mechanistic Interpretability](https://mlsanity.com/paper/2606.06333/) (2026)
- [SMIXAE: Towards Unsupervised Manifold Discovery in Language Models](https://mlsanity.com/paper/2605.09224/) (2026)
- [The Geometric Wall: Manifold Structure Predicts Layerwise Sparse Autoencoder Scaling Laws](https://mlsanity.com/paper/2605.09887/) (2026)
- [A Geometric View for Understanding Concept Learning and Neuron Interpretation in Sparse Autoencoders](https://mlsanity.com/paper/2606.07007/) (2026)
- [Critical Percolation as a Synthetic Data Model for Interpretability](https://mlsanity.com/paper/2606.20347/) (2026)
- [Geometry of Ordinal Representations in Language Models](https://mlsanity.com/paper/2607.04167/) (2026)

## Abstract (arXiv)

Sparse autoencoders (SAEs) model the activations of a neural network as linear combinations of sparsely occurring directions of variation (latents). The ability of SAEs to reconstruct activations follows scaling laws w.r.t. the number of latents. In this work, we adapt a capacity-allocation model from the neural scaling literature (Brill, 2024) to understand SAE scaling, and in particular, to understand how "feature manifolds" (multi-dimensional features) influence scaling behavior. Consistent with prior work, the model recovers distinct scaling regimes. Notably, in one regime, feature manifolds have the pathological effect of causing SAEs to learn far fewer features in data than there are latents in the SAE. We provide some preliminary discussion on whether or not SAEs are in this pathological regime in the wild.