# Mathematical Models of Computation in Superposition

arXiv: https://arxiv.org/abs/2408.05451 · HTML: https://mlsanity.com/paper/2408.05451/

## Key facts

- Cited by 35 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 15 of its 38 references that are in the map.
- First posted on arXiv 10 Aug 2024.

## Placed in

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

## Builds on (most cited first, 15 of 15)

- [Transformer Feed-Forward Layers Are Key-Value Memories](https://mlsanity.com/paper/2012.14913/) (2020)
- [Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 small](https://mlsanity.com/paper/2211.00593/) (2022)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [Towards Automated Circuit Discovery for Mechanistic Interpretability](https://mlsanity.com/paper/2304.14997/) (2023)
- [Causal Abstractions of Neural Networks](https://mlsanity.com/paper/2106.02997/) (2021)
- [Finding Neurons in a Haystack: Case Studies with Sparse Probing](https://mlsanity.com/paper/2305.01610/) (2023)
- [Improving Dictionary Learning with Gated Sparse Autoencoders](https://mlsanity.com/paper/2404.16014/) (2024)
- [Linear Algebraic Structure of Word Senses, with Applications to Polysemy](https://mlsanity.com/paper/1601.03764/) (2016)
- [Multifaceted Feature Visualization: Uncovering the Different Types of Features Learned By Each Neuron in Deep Neural Networks](https://mlsanity.com/paper/1602.03616/) (2016)
- [Polysemanticity and Capacity in Neural Networks](https://mlsanity.com/paper/2210.01892/) (2022)
- [Identifying Functionally Important Features with End-to-End Sparse Dictionary Learning](https://mlsanity.com/paper/2405.12241/) (2024)
- [Codebook Features: Sparse and Discrete Interpretability for Neural Networks](https://mlsanity.com/paper/2310.17230/) (2023)
- [Randomized Numerical Linear Algebra : A Perspective on the Field With an Eye to Software](https://mlsanity.com/paper/2302.11474/) (2023)
- [On the Computational Intractability of Exact and Approximate Dictionary Learning](https://mlsanity.com/paper/1405.6664/) (2014)
- [Spiking Hyperdimensional Network: Neuromorphic Models Integrated with Memory-Inspired Framework](https://mlsanity.com/paper/2110.00214/) (2021)

## Built on by (most cited first, 30 of 35)

- [From Flat to Hierarchical: Extracting Sparse Representations with Matching Pursuit](https://mlsanity.com/paper/2506.03093/) (2025)
- [Weight-sparse transformers have interpretable circuits](https://mlsanity.com/paper/2511.13653/) (2025)
- [Superposition Yields Robust Neural Scaling](https://mlsanity.com/paper/2505.10465/) (2025)
- [On the Complexity of Neural Computation in Superposition](https://mlsanity.com/paper/2409.15318/) (2024)
- [Stochastic Parameter Decomposition](https://mlsanity.com/paper/2506.20790/) (2025)
- [On the Limits of Sparse Autoencoders: A Theoretical Framework and Reweighted Remedy](https://mlsanity.com/paper/2506.15963/) (2025)
- [Towards Combinatorial Interpretability of Neural Computation](https://mlsanity.com/paper/2504.08842/) (2025)
- [Adversarial Attacks Leverage Interference Between Features in Superposition](https://mlsanity.com/paper/2510.11709/) (2025)
- [AudioSAE: Towards Understanding of Audio-Processing Models with Sparse AutoEncoders](https://mlsanity.com/paper/2602.05027/) (2026)
- [Superposition as Lossy Compression: Measure with Sparse Autoencoders and Connect to Adversarial Vulnerability](https://mlsanity.com/paper/2512.13568/) (2025)
- [Linear Spatial World Models Emerge in Large Language Models](https://mlsanity.com/paper/2506.02996/) (2025)
- [Characterizing stable regions in the residual stream of LLMs](https://mlsanity.com/paper/2409.17113/) (2024)
- [The Thin Line Between Comprehension and Persuasion in LLMs](https://mlsanity.com/paper/2507.01936/) (2025)
- [Evaluating Synthetic Activations composed of SAE Latents in GPT-2](https://mlsanity.com/paper/2409.15019/) (2024)
- [Putting a Face to Forgetting: Continual Learning meets Mechanistic Interpretability](https://mlsanity.com/paper/2601.22012/) (2026)
- [Expand Neurons, Not Parameters](https://mlsanity.com/paper/2510.04500/) (2025)
- [Investigating Sensitive Directions in GPT-2: An Improved Baseline and Comparative Analysis of SAEs](https://mlsanity.com/paper/2410.12555/) (2024)
- [SAFR: Neuron Redistribution for Interpretability](https://mlsanity.com/paper/2501.16374/) (2025)
- [Compressed Computation: Dense Circuits in a Toy Model of the Universal-AND Problem](https://mlsanity.com/paper/2507.09816/) (2025)
- [Feature learning is decoupled from generalization in high capacity neural networks](https://mlsanity.com/paper/2507.19680/) (2025)
- [Binary Autoencoder for Mechanistic Interpretability of Large Language Models](https://mlsanity.com/paper/2509.20997/) (2025)
- [Evidence for feature-specific error correction in LLMs](https://mlsanity.com/paper/2606.24964/) (2026)
- [Training, Reading, and Editing Legible Transformers](https://mlsanity.com/paper/2607.08946/) (2026)
- [Polysemy of Synthetic Neurons Towards a New Type of Explanatory Categorical Vector Spaces](https://mlsanity.com/paper/2505.07831/) (2025)
- [Features Emerge as Discrete States: The First Application of SAEs to 3D Representations](https://mlsanity.com/paper/2512.11263/) (2025)
- [Similarity of Neural Network Representations in Superposition](https://mlsanity.com/paper/2604.00208/) (2026)
- [Discovering Millions of Interpretable Features with Sparse Autoencoders](https://mlsanity.com/paper/2606.26620/) (2026)
- [Compressed Computation under L4 Loss is likely Computation in Superposition](https://mlsanity.com/paper/2607.04800/) (2026)
- [Scaling Interpretable Transformers with Parity Bottleneck Layers](https://mlsanity.com/paper/2607.20652/) (2026)
- [Weight-Space Mixture-of-Experts for Implicit Neural Representation Classification](https://mlsanity.com/paper/2607.29463/) (2026)

## Abstract (arXiv)

Superposition -- when a neural network represents more ``features'' than it has dimensions -- seems to pose a serious challenge to mechanistically interpreting current AI systems. Existing theory work studies representational superposition, where superposition is only used when passing information through bottlenecks. In this work, we present mathematical models of computation in superposition, where superposition is actively helpful for efficiently accomplishing the task. We first construct a task of efficiently emulating a circuit that takes the AND of the binomm2 pairs of each of m features. We construct a 1-layer MLP that uses superposition to perform this task up to varepsilon-error, where the network only requires tildeO(m^frac23) neurons, even when the input features are themselves in superposition. We generalize this construction to arbitrary sparse boolean circuits of low depth, and then construct ``error correction'' layers that allow deep fully-connected networks of width d to emulate circuits of width tildeO(d1.5) and any polynomial depth. We conclude by providing some potential applications of our work for interpreting neural networks that implement computation in superposition.