# Equivalent Linear Mappings of Large Language Models

arXiv: https://arxiv.org/abs/2505.24293 · HTML: https://mlsanity.com/paper/2505.24293/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 20 references that are in the map.
- First posted on arXiv 30 May 2025.

## Builds on (most cited first, 16 of 16)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [GLU Variants Improve Transformer](https://mlsanity.com/paper/2002.05202/) (2020)
- [Root Mean Square Layer Normalization](https://mlsanity.com/paper/1910.07467/) (2019)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Gemma Scope: Open Sparse Autoencoders Everywhere All At Once on Gemma 2](https://mlsanity.com/paper/2408.05147/) (2024)
- [In-context Vectors: Making In Context Learning More Effective and Controllable Through Latent Space Steering](https://mlsanity.com/paper/2311.06668/) (2023)
- [Understanding and Improving Layer Normalization](https://mlsanity.com/paper/1911.07013/) (2019)
- [Generalization in diffusion models arises from geometry-adaptive harmonic representations](https://mlsanity.com/paper/2310.02557/) (2023)
- [Open Problems in Mechanistic Interpretability](https://mlsanity.com/paper/2501.16496/) (2025)
- [Robust and interpretable blind image denoising via bias-free convolutional neural networks](https://mlsanity.com/paper/1906.05478/) (2019)
- [White-Box Transformers via Sparse Rate Reduction](https://mlsanity.com/paper/2306.01129/) (2023)
- [Interpreting Neural Networks through the Polytope Lens](https://mlsanity.com/paper/2211.12312/) (2022)
- [Geometric Dynamics of Signal Propagation Predict Trainability of Transformers](https://mlsanity.com/paper/2403.02579/) (2024)
- [Fast Jacobian-Vector Product for Deep Networks](https://mlsanity.com/paper/2104.00219/) (2021)

## Built on by (most cited first, 4 of 4)

- [Verbalizable Representations Form a Global Workspace in Language Models](https://mlsanity.com/paper/2607.15495/) (2026)
- [Local Linearity of LLMs Enables Activation Steering via Model-Based Linear Optimal Control](https://mlsanity.com/paper/2604.19018/) (2026)
- [As Language Models Scale, Low-order Linear Depth Dynamics Emerge](https://mlsanity.com/paper/2603.12541/) (2026)
- [From Latent Space to Jacobian Space: Measuring, Evading, and Training Against Safety-Content Accessibility](https://mlsanity.com/paper/2610.04316/) (2026)

## Abstract (arXiv)

Despite significant progress in transformer interpretability, an understanding of the computational mechanisms of large language models (LLMs) remains a fundamental challenge. Many approaches interpret a network's hidden representations but remain agnostic about how those representations are generated. We address this by mapping LLM inference for a given input sequence to an equivalent and interpretable linear system which reconstructs the predicted output embedding with relative error below 10-13 at double floating-point precision, requiring no additional model training. We exploit a property of transformers wherein every operation (gated activations, attention, and normalization) can be expressed as A(x) · x, where A(x) represents an input-dependent linear transform and x preserves the linear pathway. To expose this linear structure, we strategically detach components of the gradient computation with respect to an input sequence, freezing the A(x) terms at their values computed during inference, such that the Jacobian yields an equivalent linear mapping. This detached Jacobian of the model reconstructs the output with one linear operator per input token, which is shown for Qwen 3, Gemma 3 and Llama 3, up to Qwen 3 14B. These linear representations demonstrate that LLMs operate in extremely low-dimensional subspaces where the singular vectors can be decoded to interpretable semantic concepts. The computation for each intermediate output also has a linear equivalent, and we examine how the linear representations of individual layers and their attention and multilayer perceptron modules build predictions, and use these as steering operators to insert semantic concepts into unrelated text. Despite their global nonlinearity, LLMs can be interpreted through equivalent linear representations that reveal low-dimensional semantic structures in the next-token prediction process.