半监督Hopfield模型：理论与数值结果
Semi-supervised Hopfield model: Theoretical and Numerical results
AI总结：
本文针对Hopfield网络提出半监督Hebbian学习的统计力学理论，通过凸组合监督与无监督Hebbian核构建模型，经理论推导和模拟验证，发现混合策略性能更优，λ为学习超参数。
AI中文摘要：
在机器学习的日常实践中，全标注数据集是一种奢侈：标注需要昂贵且耗时的人工注释，而原始未标注数据可自动批量获取。半监督学习是应对这种不对称性的标准方案，即网络联合利用少量标注样本和大量未标注样本，但目前仍缺乏半监督Hebbian学习的统计力学理论。本文针对Hopfield网络填补了这一空白：我们规定突触耦合为监督和无监督Hebbian核的凸组合，权重为[0,1]区间内的混合参数λ，二者构建于相同原型之上，我们求解所得网络的涌现计算能力。信噪比分析得出一步Mattis磁化强度和学习阈值，即稳定检索所需的最小数据集规模。利用Guerra插值，我们在高存储 regime下推导了复制对称淬火压力，通过特定本征通道分解处理监督和无监督通道产生的相关无序。所得相图显示混合策略优于两种纯协议。最后，我们证明淬火压力关于λ是凸的，因此热力学无法选择内部混合：λ因此是一个学习超参数。所有分析结果均通过广泛的蒙特卡洛模拟得到验证。
英文摘要：
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.
