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

ICLR 2026 Conference Submission20998 Authors

Keywords: Implicit Bias, Invariance, Hopfield Networks

TL;DR: Classical Hopfield networks can generalize from a few graphs to memorize an entire isomorphism class by finding approximately invariant solutions.

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.

Primary Area: learning theory

Submission Number: 20998