Semi-supervised Hopfield model: Theoretical and Numerical results
arXiv:2607.28173
The paper develops a statistical‑mechanical theory for semi‑supervised learning in Hopfield networks by mixing supervised and unsupervised Hebbian couplings, deriving performance thresholds and phase diagrams, and confirming results with simulations.
Abstract
In the daily practice of Machine Learning, fully labeled datasets are a luxury: labels demand expensive and time-consuming human annotation, whereas raw, unlabeled data can be harvested automatically and in bulk. Semi-supervised learning, where the network jointly exploits the few labeled and the many unlabeled examples at its disposal, is the standard answer to this asymmetry, yet a statistical mechanical theory of semi-supervised Hebbian learning is still lacking. In this paper we fill this gap for the Hopfield network: we prescribe a synaptic coupling given by the convex combination, weighted by a mixing parameter λin [0,1], of the supervised and unsupervised Hebbian kernels built from the same archetypes, and we solve for the emergent computational capabilities of the resulting network. A signal-to-noise analysis yields the one-step Mattis magnetization and the learning threshold, i.e. the minimum dataset size for stable retrieval. Using Guerra's interpolation, we then derive the Replica Symmetric quenched pressure in the high-storage regime, treating the correlated disorder generated by the supervised and unsupervised channels through a particular eigen-channel decomposition. The resulting phase diagram shows that a mixed strategy outperforms both pure protocols. Finally, we prove that the quenched pressure is convex in λ, so thermodynamics cannot select an interior mixture: λis therefore a learning hyperparameter. All the analytical findings are successfully checked against extensive Monte Carlo simulations.