# Bilinear autoencoders find interpretable manifolds

arXiv: https://arxiv.org/abs/2605.08891 · HTML: https://mlsanity.com/paper/2605.08891/

## Key facts

- Cited by 1 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 22 of its 45 references that are in the map.
- First posted on arXiv 9 May 2026.

## Placed in

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

## Builds on (most cited first, 22 of 22)

- [GLU Variants Improve Transformer](https://mlsanity.com/paper/2002.05202/) (2020)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale](https://mlsanity.com/paper/2406.17557/) (2024)
- [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)
- [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)
- [Sparse Autoencoders Trained on the Same Data Learn Different Features](https://mlsanity.com/paper/2501.16615/) (2025)
- [Sparse Autoencoders Do Not Find Canonical Units of Analysis](https://mlsanity.com/paper/2502.04878/) (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)
- [The Non-Linear Representation Dilemma: Is Causal Abstraction Enough for Mechanistic Interpretability?](https://mlsanity.com/paper/2507.08802/) (2025)
- [Everything, Everywhere, All at Once: Is Mechanistic Interpretability Identifiable?](https://mlsanity.com/paper/2502.20914/) (2025)
- [Into the Rabbit Hull: From Task-Relevant Concepts in DINO to Minkowski Geometry](https://mlsanity.com/paper/2510.08638/) (2025)
- [Do Sparse Autoencoders Capture Concept Manifolds?](https://mlsanity.com/paper/2604.28119/) (2026)
- [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)
- [Position: Mechanistic Interpretability Should Prioritize Feature Consistency in SAEs](https://mlsanity.com/paper/2505.20254/) (2025)
- [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)
- [From Atoms to Trees: Building a Structured Feature Forest with Hierarchical Sparse Autoencoders](https://mlsanity.com/paper/2602.11881/) (2026)

## Built on by (most cited first, 1 of 1)

- [Activation-Space Order-Swap Geometry: A Site-Asymmetry Audit](https://mlsanity.com/paper/2608.25315/) (2026)

## Abstract (arXiv)

Sparse autoencoders have become a standard tool for uncovering interpretable latent representations in neural networks. Yet salient concepts often span manifolds that current linear methods cannot capture without post hoc analysis. This paper uses quadratic latents to close this gap: we implement these with bilinear autoencoders, which decompose activations into low-rank quadratic forms, compose linearly in weight space, and admit input-independent geometric analysis. This qualitative difference in what concepts quadratic latents can detect challenges the standard linear representation hypothesis. Our experiments and visualisations show that multi-dimensional geometries are highly prevalent and that composite latents capture them well, systematically improving reconstruction error in language models. Furthermore, we show that autoencoders with varying geometric priors recover the same input subspace despite their dictionary entries being distinct. Practically, these models serve as an unsupervised tool for manifold discovery, which we demonstrate through an interactive online visualizer for Qwen 3.5. This is a step toward nonlinear but mathematically tractable latent representations whose composition is expressive and interpretable by design.