mlsanityData to 14 Oct 2026

Tensor Product Representation Probes Reveal Shared Structure Across Linear Directions

arXiv 2605.09967 ↗First posted 11 May 2026cs.LGcited by 1

On the map: Mechanistic interpretability → Sparse autoencoder training and evaluation

Abstract (arXiv)

While researchers are finding concepts represented as linear directions in language models, a bag of linear directions fails to capture relational structure. To better understand this dichotomy, we study a model with known linear representations, but trained in a highly structured domain -- the board game Othello. While the model's internal board-state representation is linearly decodable, we find additional structure in the form of tensor product representations (TPRs). We train TPR probes to recover shared structure amongst the linear probes, yielding a factorization into square-embeddings, color-embeddings, and a binding matrix that composes them to construct the model's board-state representation. We find geometric signatures within the weights of our TPR probe that align with the structure of the board, but perhaps more importantly, that the linear probes can be recovered directly from the parameters of our TPR probe. Our findings suggest that directional representations may be projections of more structured underlying representations.

Key facts. Cited by 1 paper in the map (data through 14 Oct 2026). Placed in Sparse autoencoder training and evaluation, one of 13 directions in Mechanistic interpretability. Builds on 32 of its 37 references that are in the map. First posted on arXiv 11 May 2026.

Builds on

32 of 37 references.

Closest ancestors · 4

Same direction, newest first

  1. 2025From Flat to Hierarchical: Extracting Sparse Representations with Matching Pursuit3 Jun
  2. 2025The Origins of Representation Manifolds in Large Language Models23 May
  3. 2025Learning Multi-Level Features with Matryoshka Sparse Autoencoders21 Mar
  4. 2025Projecting Assumptions: The Duality Between Sparse Autoencoders and Concept Geometry3 Mar

Other references · 28

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  3. 2022Emergent World Representations: Exploring a Sequence Model Trained on a Synthetic Taskcited by 431
  4. 2023Emergent Linear Representations in World Models of Self-Supervised Sequence Modelscited by 319
  5. 2024Not All Language Model Features Are One-Dimensionally Linearcited by 212
  6. 2024A Mechanistic Understanding of Alignment Algorithms: A Case Study on DPO and Toxicitycited by 185
  7. 2023Linear Representations of Sentiment in Large Language Modelscited by 163
  8. 2025Language Models Use Trigonometry to Do Additioncited by 70
  9. 2024ICLR: In-Context Learning of Representationscited by 50
  10. 2024Designing a Dashboard for Transparency and Control of Conversational AIcited by 46
  11. 2026When Models Manipulate Manifolds: The Geometry of a Counting Taskcited by 46
  12. 2025Shared Global and Local Geometry of Language Model Embeddingscited by 36
16 more
  1. 2019Enhancing the Transformer with Explicit Relational Encoding for Math Problem Solvingcited by 35
  2. 2018Learning to Reason with Third-Order Tensor Productscited by 27
  3. 2018RNNs Implicitly Implement Tensor Product Representationscited by 26
  4. 2024Relational Composition in Neural Networks: A Survey and Call to Actioncited by 23
  5. 2025Temporal Sparse Autoencoders: Leveraging the Sequential Nature of Language for Interpretabilitycited by 18
  6. 2026Valence-Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Controlcited by 16
  7. 2026The Shape of Beliefs: Geometry, Dynamics, and Interventions along Representation Manifolds of Language Models' Posteriorscited by 15
  8. 2025Incorporating Hierarchical Semantics in Sparse Autoencoder Architecturescited by 14
  9. 2025The Geometry of Self-Verification in a Task-Specific Reasoning Modelcited by 13
  10. 2021Enriching Transformers with Structured Tensor-Product Representations for Abstractive Summarizationcited by 11
  11. 2026Learning a Generative Meta-Model of LLM Activationscited by 8
  12. 2025Why Can't Transformers Learn Multiplication? Reverse-Engineering Reveals Long-Range Dependency Pitfallscited by 7
  13. 2025From Isolation to Entanglement: When Do Interpretability Methods Identify and Disentangle Known Concepts?cited by 7
  14. 2026Transformers learn factored representationscited by 7
  15. 2026Decomposing Query-Key Feature Interactions Using Contrastive Covariancescited by 3
  16. 2024Attention-based Iterative Decomposition for Tensor Product Representationcited by 2

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 1.

1 paper.

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