# Provably Extracting the Features from a General Superposition

arXiv: https://arxiv.org/abs/2512.15987 · HTML: https://mlsanity.com/paper/2512.15987/

## 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 35 of its 51 references that are in the map.
- First posted on arXiv 17 Dec 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 35)

- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [Stealing Machine Learning Models via Prediction APIs](https://mlsanity.com/paper/1609.02943/) (2016)
- [Neural Additive Models: Interpretable Machine Learning with Neural Nets](https://mlsanity.com/paper/2004.13912/) (2020)
- [Stealing Part of a Production Language Model](https://mlsanity.com/paper/2403.06634/) (2024)
- [Recovery Guarantees for One-hidden-layer Neural Networks](https://mlsanity.com/paper/1706.03175/) (2017)
- [Learning Single-Index Models with Shallow Neural Networks](https://mlsanity.com/paper/2210.15651/) (2022)
- [Reverse-Engineering Deep ReLU Networks](https://mlsanity.com/paper/1910.00744/) (2019)
- [Learning One-hidden-layer Neural Networks with Landscape Design](https://mlsanity.com/paper/1711.00501/) (2017)
- [Smoothing the Landscape Boosts the Signal for SGD: Optimal Sample Complexity for Learning Single Index Models](https://mlsanity.com/paper/2305.10633/) (2023)
- [Superpolynomial Lower Bounds for Learning One-Layer Neural Networks using Gradient Descent](https://mlsanity.com/paper/2006.12011/) (2020)
- [On Learning Gaussian Multi-index Models with Gradient Flow](https://mlsanity.com/paper/2310.19793/) (2023)
- [Statistical-Query Lower Bounds via Functional Gradients](https://mlsanity.com/paper/2006.15812/) (2020)
- [Algorithms and SQ Lower Bounds for PAC Learning One-Hidden-Layer ReLU Networks](https://mlsanity.com/paper/2006.12476/) (2020)
- [Computational-Statistical Gaps in Gaussian Single-Index Models](https://mlsanity.com/paper/2403.05529/) (2024)
- [Logits of API-Protected LLMs Leak Proprietary Information](https://mlsanity.com/paper/2403.09539/) (2024)
- [Learning Two-layer Neural Networks with Symmetric Inputs](https://mlsanity.com/paper/1810.06793/) (2018)
- [On the Cryptographic Hardness of Learning Single Periodic Neurons](https://mlsanity.com/paper/2106.10744/) (2021)
- [Agnostically Learning Single-Index Models using Omnipredictors](https://mlsanity.com/paper/2306.10615/) (2023)
- [Efficient Algorithms for Learning Depth-2 Neural Networks with General ReLU Activations](https://mlsanity.com/paper/2107.10209/) (2021)
- [Learning Multi-Index Models with Neural Networks via Mean-Field Langevin Dynamics](https://mlsanity.com/paper/2408.07254/) (2024)
- [Survey on Algorithms for multi-index models](https://mlsanity.com/paper/2504.05426/) (2025)
- [Non-Gaussian Component Analysis via Lattice Basis Reduction](https://mlsanity.com/paper/2112.09104/) (2021)
- [Classification Under Misspecification: Halfspaces, Generalized Linear Models, and Connections to Evolvability](https://mlsanity.com/paper/2006.04787/) (2020)
- [Learning sum of diverse features: computational hardness and efficient gradient-based training for ridge combinations](https://mlsanity.com/paper/2406.11828/) (2024)
- [Learning Hidden Markov Models Using Conditional Samples](https://mlsanity.com/paper/2302.14753/) (2023)
- [An Exact Poly-Time Membership-Queries Algorithm for Extraction a three-Layer ReLU Network](https://mlsanity.com/paper/2105.09673/) (2021)
- [Robustly Learning Single-Index Models via Alignment Sharpness](https://mlsanity.com/paper/2402.17756/) (2024)
- [Model Stealing for Any Low-Rank Language Model](https://mlsanity.com/paper/2411.07536/) (2024)

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

- [Size Doesn't Matter: Cosine-Scored Sparse Autoencoders](https://mlsanity.com/paper/2606.15054/) (2026)

## Abstract (arXiv)

It is widely believed that complex machine learning models generally encode features through linear representations. This is the foundational hypothesis behind a vast body of work on interpretability. A key challenge toward extracting interpretable features, however, is that they exist in superposition. In this work, we study the question of extracting features in superposition from a learning theoretic perspective. We start with the following fundamental setting: we are given query access to a function [ f(x)=sumi=1n σi(vi^top x), ] where each unit vector vi encodes a feature direction and σi:R→R is an arbitrary response function and our goal is to recover the vi and the function f. In learning-theoretic terms, superposition refers to the overcomplete regime, when the number of features is larger than the underlying dimension (i.e. n > d), which has proven especially challenging for typical algorithmic approaches. Our main result is an efficient query algorithm that, from noisy oracle access to f, identifies all feature directions whose responses are non-degenerate and reconstructs the function f. Crucially, our algorithm works in a significantly more general setting than all related prior results. We allow for essentially arbitrary superpositions, only requiring that vi, vj are not nearly identical for i neq j, and allowing for general response functions σi. At a high level, our algorithm introduces an approach for searching in Fourier space by iteratively refining the search space to locate the hidden directions vi.