Implicit Bias and Invariance: How Hopfield Networks Efficiently Learn Graph Orbits

• URL: http://arxiv.org/abs/2512.14338v1
• Date: Tue, 16 Dec 2025 12:06:58 GMT
• Title: Implicit Bias and Invariance: How Hopfield Networks Efficiently Learn Graph Orbits
• Authors: Michael Murray, Tenzin Chan, Kedar Karhadker, Christopher J. Hillar,
• Abstract summary: We infer the full isomorphism class of a graph from a small random sample.<n>Findings highlight a unifying mechanism for generalization in Hopfield networks.
• Score: 9.02293509509624
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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