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

Excerpt

arXiv:2512.14338v4 Announce Type: replace Abstract: Many learning problems are organized by group symmetries. While invariance is often imposed through architectures or group averaging, we ask when it can emerge from training on a finite random subset of an orbit. We study this question in classical Hopfield networks, where strict memorization can be expressed as a linear margin problem. Reparameterizing minimization of energy flow (MEF) as an exponential loss connects gradient descent to the co