# Finding Manifolds With Bilinear Autoencoders

arXiv: https://arxiv.org/abs/2510.16820 · HTML: https://mlsanity.com/paper/2510.16820/

## Key facts

- Cited by 5 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 23 of its 34 references that are in the map.
- First posted on arXiv 19 Oct 2025.

## Placed in

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

## Builds on (most cited first, 23 of 23)

- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [GLU Variants Improve Transformer](https://mlsanity.com/paper/2002.05202/) (2020)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [Language Modeling with Gated Convolutional Networks](https://mlsanity.com/paper/1612.08083/) (2016)
- [Scaling and evaluating sparse autoencoders](https://mlsanity.com/paper/2406.04093/) (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)
- [A is for Absorption: Studying Feature Splitting and Absorption in Sparse Autoencoders](https://mlsanity.com/paper/2409.14507/) (2024)
- [Learning Multi-Level Features with Matryoshka Sparse Autoencoders](https://mlsanity.com/paper/2503.17547/) (2025)
- [BatchTopK Sparse Autoencoders](https://mlsanity.com/paper/2412.06410/) (2024)
- [Sparse Autoencoders Trained on the Same Data Learn Different Features](https://mlsanity.com/paper/2501.16615/) (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)
- [Everything, Everywhere, All at Once: Is Mechanistic Interpretability Identifiable?](https://mlsanity.com/paper/2502.20914/) (2025)
- [Bilinear MLPs enable weight-based mechanistic interpretability](https://mlsanity.com/paper/2410.08417/) (2024)
- [Interpretability as Compression: Reconsidering SAE Explanations of Neural Activations with MDL-SAEs](https://mlsanity.com/paper/2410.11179/) (2024)
- [Dense SAE Latents Are Features, Not Bugs](https://mlsanity.com/paper/2506.15679/) (2025)
- [Towards Compositional Interpretability for XAI](https://mlsanity.com/paper/2406.17583/) (2024)
- [Compositionality as we see it, everywhere around us](https://mlsanity.com/paper/2110.05327/) (2021)
- [Compositionality Unlocks Deep Interpretable Models](https://mlsanity.com/paper/2504.02667/) (2025)

## Built on by (most cited first, 5 of 5)

- [Do Sparse Autoencoders Capture Concept Manifolds?](https://mlsanity.com/paper/2604.28119/) (2026)
- [The Information Geometry of Softmax: Probing and Steering](https://mlsanity.com/paper/2602.15293/) (2026)
- [PolySAE: Modeling Feature Interactions in Sparse Autoencoders via Polynomial Decoding](https://mlsanity.com/paper/2602.01322/) (2026)
- [SMIXAE: Towards Unsupervised Manifold Discovery in Language Models](https://mlsanity.com/paper/2605.09224/) (2026)
- [WriteSAE: Sparse Autoencoders for Recurrent State](https://mlsanity.com/paper/2605.12770/) (2026)

## Abstract (arXiv)

Sparse autoencoders are a standard tool for uncovering interpretable latent representations in neural networks. Yet, their interpretation depends on the inputs, making their isolated study incomplete. Polynomials offer a solution; they serve as algebraic primitives that can be analysed without reference to input and can describe structures ranging from linear concepts to complicated manifolds. This work uses bilinear autoencoders to efficiently decompose representations into quadratic polynomials. We discuss improvements that induce importance ordering, clustering, and activation sparsity. This is an initial step toward nonlinear yet analysable latents through their algebraic properties.