{"ID":2828481,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.14338","arxiv_id":"2512.14338","title":"Implicit Bias and Invariance: How Hopfield Networks Efficiently Learn Graph Orbits","abstract":"Many learning problems involve symmetries, and while invariance can be built into neural architectures, it can also emerge implicitly when training on group-structured data. We study this phenomenon in classical Hopfield networks and show they can infer the full isomorphism class of a graph from a small random sample. Our results reveal that: (i) graph isomorphism classes can be represented within a three-dimensional invariant subspace, (ii) using gradient descent to minimize energy flow (MEF) has an implicit bias toward norm-efficient solutions, which underpins a polynomial sample complexity bound for learning isomorphism classes, and (iii) across multiple learning rules, parameters converge toward the invariant subspace as sample sizes grow. Together, these findings highlight a unifying mechanism for generalization in Hopfield networks: a bias toward norm efficiency in learning drives the emergence of approximate invariance under group-structured data.","short_abstract":"Many learning problems involve symmetries, and while invariance can be built into neural architectures, it can also emerge implicitly when training on group-structured data. We study this phenomenon in classical Hopfield networks and show they can infer the full isomorphism class of a graph from a small random sample....","url_abs":"https://arxiv.org/abs/2512.14338","url_pdf":"https://arxiv.org/pdf/2512.14338v2","authors":"[\"Michael Murray\",\"Tenzin Chan\",\"Kedar Karhadker\",\"Christopher J. Hillar\"]","published":"2025-12-16T12:06:58Z","proceeding":"cs.LG","tasks":"[\"cs.LG\"]","methods":"[]","has_code":false}