Equivalent Linear Mappings of Large Language Models

James Robert Golden

Open Source Links: https://github.com/jamesgolden1/equivalent-linear-LLMs/

Keywords: Foundational work, Steering

Other Keywords: Interpretability, Jacobian, Linear

TL;DR: LLMs like Qwen 3 14B and Gemma 3 12B have exactly equivalent linear representations for next-token prediction on any given input text, which can be used to interpret emergence of predictions and as a steering operator.

Abstract: 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 transformer decoders wherein every operation (gated activations, attention, and normalization) can be expressed as $A(x) \cdot x$, 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 expressive power and global nonlinearity, modern LLMs can be interpreted through equivalent linear representations that reveal low-dimensional semantic structures in the next-token prediction process. Code is available at \url{https://github.com/jamesgolden1/equivalent-linear-LLMs/ } .

Submission Number: 57