Geometric Factual Recall in Transformers
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arXiv
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| Format: | Preprint |
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2026
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| author | Ravfogel, Shauli Yehudai, Gilad Bruna, Joan Bietti, Alberto |
| author_facet | Ravfogel, Shauli Yehudai, Gilad Bruna, Joan Bietti, Alberto |
| contents | 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, \emph{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 \emph{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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12426 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometric Factual Recall in Transformers Ravfogel, Shauli Yehudai, Gilad Bruna, Joan Bietti, Alberto Computation and Language 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, \emph{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 \emph{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. |
| title | Geometric Factual Recall in Transformers |
| topic | Computation and Language |
| url | https://arxiv.org/abs/2605.12426 |