# Weight-based Decomposition: A Case for Bilinear MLPs

arXiv: https://arxiv.org/abs/2406.03947 · HTML: https://mlsanity.com/paper/2406.03947/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 25 references that are in the map.
- First posted on arXiv 6 Jun 2024.

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

- [Google's Neural Machine Translation System: Bridging the Gap between Human and Machine Translation](https://mlsanity.com/paper/1609.08144/) (2016)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [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)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Language Modeling with Gated Convolutional Networks](https://mlsanity.com/paper/1612.08083/) (2016)
- [Sparse Feature Circuits: Discovering and Editing Interpretable Causal Graphs in Language Models](https://mlsanity.com/paper/2403.19647/) (2024)
- [A mathematical theory of semantic development in deep neural networks](https://mlsanity.com/paper/1810.10531/) (2018)
- [A Toy Model of Universality: Reverse Engineering How Networks Learn Group Operations](https://mlsanity.com/paper/2302.03025/) (2023)
- [Interpreting Neural Networks through the Polytope Lens](https://mlsanity.com/paper/2211.12312/) (2022)
- [Opening the AI black box: program synthesis via mechanistic interpretability](https://mlsanity.com/paper/2402.05110/) (2024)
- [Using Degeneracy in the Loss Landscape for Mechanistic Interpretability](https://mlsanity.com/paper/2405.10927/) (2024)
- [A technical note on bilinear layers for interpretability](https://mlsanity.com/paper/2305.03452/) (2023)
- [The Local Interaction Basis: Identifying Computationally-Relevant and Sparsely Interacting Features in Neural Networks](https://mlsanity.com/paper/2405.10928/) (2024)

## Built on by (most cited first, 2 of 2)

- [AERO: Entropy-Guided Framework for Private LLM Inference](https://mlsanity.com/paper/2410.13060/) (2024)
- [Explicit Fuzzy Logic in the Feed-Forward Layer: Self-Forgetting Quantifiers Discover Legible Grammatical-Licensing Detectors](https://mlsanity.com/paper/2606.31845/) (2026)

## Abstract (arXiv)

Gated Linear Units (GLUs) have become a common building block in modern foundation models. Bilinear layers drop the non-linearity in the "gate" but still have comparable performance to other GLUs. An attractive quality of bilinear layers is that they can be fully expressed in terms of a third-order tensor and linear operations. Leveraging this, we develop a method to decompose the bilinear tensor into a set of sparsely interacting eigenvectors that show promising interpretability properties in preliminary experiments for shallow image classifiers (MNIST) and small language models (Tiny Stories). Since the decomposition is fully equivalent to the model's original computations, bilinear layers may be an interpretability-friendly architecture that helps connect features to the model weights. Application of our method may not be limited to pretrained bilinear models since we find that language models such as TinyLlama-1.1B can be finetuned into bilinear variants.