# Geometric Factual Recall in Transformers

arXiv: https://arxiv.org/abs/2605.12426 · HTML: https://mlsanity.com/paper/2605.12426/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 40 references that are in the map.
- First posted on arXiv 12 May 2026.

## Builds on (most cited first, 30 of 36)

- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [Language Models as Knowledge Bases?](https://mlsanity.com/paper/1909.01066/) (2019)
- [Transformer Feed-Forward Layers Are Key-Value Memories](https://mlsanity.com/paper/2012.14913/) (2020)
- [Mass-Editing Memory in a Transformer](https://mlsanity.com/paper/2210.07229/) (2022)
- [Embedding Entities and Relations for Learning and Inference in Knowledge Bases](https://mlsanity.com/paper/1412.6575/) (2014)
- [Complex Embeddings for Simple Link Prediction](https://mlsanity.com/paper/1606.06357/) (2016)
- [How Much Knowledge Can You Pack Into the Parameters of a Language Model?](https://mlsanity.com/paper/2002.08910/) (2020)
- [Knowledge Neurons in Pretrained Transformers](https://mlsanity.com/paper/2104.08696/) (2021)
- [Dissecting Recall of Factual Associations in Auto-Regressive Language Models](https://mlsanity.com/paper/2304.14767/) (2023)
- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Are Transformers universal approximators of sequence-to-sequence functions?](https://mlsanity.com/paper/1912.10077/) (2019)
- [Towards Revealing the Mystery behind Chain of Thought: A Theoretical Perspective](https://mlsanity.com/paper/2305.15408/) (2023)
- [Physics of Language Models: Part 3.1, Knowledge Storage and Extraction](https://mlsanity.com/paper/2309.14316/) (2023)
- [Chain of Thought Empowers Transformers to Solve Inherently Serial Problems](https://mlsanity.com/paper/2402.12875/) (2024)
- [Dense Associative Memory for Pattern Recognition](https://mlsanity.com/paper/1606.01164/) (2016)
- [Linearity of Relation Decoding in Transformer Language Models](https://mlsanity.com/paper/2308.09124/) (2023)
- [Do Large Language Models Latently Perform Multi-Hop Reasoning?](https://mlsanity.com/paper/2402.16837/) (2024)
- [Birth of a Transformer: A Memory Viewpoint](https://mlsanity.com/paper/2306.00802/) (2023)
- [Canonical Tensor Decomposition for Knowledge Base Completion](https://mlsanity.com/paper/1806.07297/) (2018)
- [Physics of Language Models: Part 3.3, Knowledge Capacity Scaling Laws](https://mlsanity.com/paper/2404.05405/) (2024)
- [Physics of Language Models: Part 3.2, Knowledge Manipulation](https://mlsanity.com/paper/2309.14402/) (2023)
- [Representational Strengths and Limitations of Transformers](https://mlsanity.com/paper/2306.02896/) (2023)
- [Conditional Memory via Scalable Lookup: A New Axis of Sparsity for Large Language Models](https://mlsanity.com/paper/2601.07372/) (2026)
- [Knowledge Graph Completion via Complex Tensor Factorization](https://mlsanity.com/paper/1702.06879/) (2017)
- [Hopping Too Late: Exploring the Limitations of Large Language Models on Multi-Hop Queries](https://mlsanity.com/paper/2406.12775/) (2024)
- [Understanding Factual Recall in Transformers via Associative Memories](https://mlsanity.com/paper/2412.06538/) (2024)
- [Are Transformers with One Layer Self-Attention Using Low-Rank Weight Matrices Universal Approximators?](https://mlsanity.com/paper/2307.14023/) (2023)
- [On the Optimal Memorization Power of ReLU Neural Networks](https://mlsanity.com/paper/2110.03187/) (2021)
- [Memorization Capacity of Multi-Head Attention in Transformers](https://mlsanity.com/paper/2306.02010/) (2023)
- [Scaling Laws for Associative Memories](https://mlsanity.com/paper/2310.02984/) (2023)

## Built on by (most cited first, 1 of 1)

- [Fine-Tuning Dynamics of In-Context Factual Recall in Transformers](https://mlsanity.com/paper/2605.27774/) (2026)

## Abstract (arXiv)

How do transformer language models memorize factual associations? A common view casts internal weight matrices as associative memories over pairs of embeddings, requiring parameter counts that scale linearly with the number of facts. We develop a theoretical and empirical account of an alternative, geometric form of memorization in which learned embeddings encode relational structure directly, and the MLP plays a qualitatively different role. In a controlled setting where a single-layer transformer must memorize random bijections from subjects to a shared attribute set, we prove that a logarithmic embedding dimension suffices: subject embeddings encode linear superpositions of their associated attribute vectors, and a small MLP acts as a relation-conditioned selector that extracts the relevant attribute via ReLU gating, and not as an associative key-value mapping. We extend these results to the multi-hop setting -- chains of relational queries such as ``Who is the mother of the wife of x?'' -- providing constructions with and without chain-of-thought that exhibit a provable capacity-depth tradeoff, complemented by a matching information-theoretic lower bound. Empirically, gradient descent discovers solutions with precisely the predicted structure. Once trained, the MLP transfers zero-shot to entirely new bijections when subject embeddings are appropriately re-initialized, revealing that it has learned a generic selection mechanism rather than memorized any particular set of facts.