# Transformers learn factored representations

arXiv: https://arxiv.org/abs/2602.02385 · HTML: https://mlsanity.com/paper/2602.02385/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 34 references that are in the map.
- First posted on arXiv 2 Feb 2026.

## Builds on (most cited first, 19 of 19)

- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [The Platonic Representation Hypothesis](https://mlsanity.com/paper/2405.07987/) (2024)
- [Transformers Learn Shortcuts to Automata](https://mlsanity.com/paper/2210.10749/) (2022)
- [Not All Language Model Features Are One-Dimensionally Linear](https://mlsanity.com/paper/2405.14860/) (2024)
- [Getting aligned on representational alignment](https://mlsanity.com/paper/2310.13018/) (2023)
- [Causal Machine Learning: A Survey and Open Problems](https://mlsanity.com/paper/2206.15475/) (2022)
- [Transformers represent belief state geometry in their residual stream](https://mlsanity.com/paper/2405.15943/) (2024)
- [Language Models Use Trigonometry to Do Addition](https://mlsanity.com/paper/2502.00873/) (2025)
- [When Models Manipulate Manifolds: The Geometry of a Counting Task](https://mlsanity.com/paper/2601.04480/) (2026)
- [Questioning Representational Optimism in Deep Learning: The Fractured Entangled Representation Hypothesis](https://mlsanity.com/paper/2505.11581/) (2025)
- [Dense SAE Latents Are Features, Not Bugs](https://mlsanity.com/paper/2506.15679/) (2025)
- [Disentangling Representations of Text by Masking Transformers](https://mlsanity.com/paper/2104.07155/) (2021)
- [Constrained belief updates explain geometric structures in transformer representations](https://mlsanity.com/paper/2502.01954/) (2025)
- [Nearly Maximally Predictive Features and Their Dimensions](https://mlsanity.com/paper/1702.08565/) (2017)
- [Neural networks leverage nominally quantum and post-quantum representations](https://mlsanity.com/paper/2507.07432/) (2025)
- [FACTS: A Factored State-Space Framework For World Modelling](https://mlsanity.com/paper/2410.20922/) (2024)
- [Vector Arithmetic in Concept and Token Subspaces](https://mlsanity.com/paper/2511.18162/) (2025)
- [From monoliths to modules: Decomposing transducers for efficient world modelling](https://mlsanity.com/paper/2512.02193/) (2025)
- [Identifiability and minimality bounds of quantum and post-quantum models of classical stochastic processes](https://mlsanity.com/paper/2509.03004/) (2025)

## Built on by (most cited first, 7 of 7)

- [Arithmetic in the Wild: Llama uses Base-10 Addition to Reason About Cyclic Concepts](https://mlsanity.com/paper/2605.01148/) (2026)
- [Task Vector Geometry Underlies Dual Modes of Task Inference in Transformers](https://mlsanity.com/paper/2605.03780/) (2026)
- [Tensor Product Representation Probes Reveal Shared Structure Across Linear Directions](https://mlsanity.com/paper/2605.09967/) (2026)
- [Radical AI Interpretability](https://mlsanity.com/paper/2606.26523/) (2026)
- [World models of environment, agent and joint agent-environment systems](https://mlsanity.com/paper/2608.20401/) (2026)
- [Approximate Homomorphisms and Convergent Representations in Transducers](https://mlsanity.com/paper/2608.20428/) (2026)
- [LLMs program using Reusable and Composable Abstract Representations](https://mlsanity.com/paper/2610.19238/) (2026)

## Abstract (arXiv)

Transformers pretrained via next token prediction learn to factor their world into parts, representing these factors in orthogonal subspaces of the residual stream. We formalize two representational hypotheses: (1) a representation in the product space of all factors, whose dimension grows exponentially with the number of parts, or (2) a factored representation in orthogonal subspaces, whose dimension grows linearly. The factored representation is lossless when factors are conditionally independent, but sacrifices predictive fidelity otherwise, creating a tradeoff between dimensional efficiency and accuracy. We derive precise predictions about the geometric structure of activations for each, including the number of subspaces, their dimensionality, and the arrangement of context embeddings within them. We test between these hypotheses on transformers trained on synthetic processes with known latent structure. Models learn factored representations when factors are conditionally independent, and continue to favor them early in training even when noise or hidden dependencies undermine conditional independence, reflecting an inductive bias toward factoring at the cost of fidelity. This provides a principled explanation for why transformers decompose the world into parts, and suggests that interpretable low dimensional structure may persist even in models trained on complex data.