Dense Hopfield networks are known for their feature to prototype transition and adversarial robustness. However, previous theoretical studies have been mostly concerned with their storage capacity. We bridge this gap by studying the phase diagram of p-body Hopfield networks in the teacher-student setting of an unsupervised learning problem, uncovering ferromagnetic phases reminiscent of the prototype and feature learning regimes. On the Nishimori line, we find the critical size of the training set necessary for efficient pattern retrieval. Interestingly, we find that that the paramagnetic to ferromagnetic transition of the teacher-student setting coincides with the paramagnetic to spin-glass transition of the direct model, i.e. with random patterns. Outside of the Nishimori line, we investigate the learning performance in relation to the inference temperature and dataset noise. Moreover, we show that using a larger p for the student than the teacher gives the student an extensive tolerance to noise. We then derive a closed-form expression measuring the adversarial robustness of such a student at zero temperature, corroborating the positive correlation between number of parameters and robustness observed in large neural networks. We also use our model to clarify why the prototype phase of modern Hopfield networks is adversarially robust.

翻译：密集Hopfield网络以其特征到原型的转变特性及对抗鲁棒性而著称。然而，先前的理论研究主要关注其存储容量。我们通过研究无监督学习问题师生设置中p体Hopfield网络的相图来填补这一空白，发现了令人联想到原型学习机制与特征学习机制的铁磁相。在西岛直线上，我们找到了实现高效模式检索所需训练集的临界规模。有趣的是，我们发现师生设置中的顺磁-铁磁转变与直接模型（即随机模式）的顺磁-自旋玻璃转变相重合。在西岛直线之外，我们探究了学习性能与推理温度及数据集噪声的关系。此外，我们证明当学生网络采用比教师网络更大的p值时，学生网络会获得对噪声的广延容忍度。随后我们推导出零温度下衡量此类学生网络对抗鲁棒性的闭式表达式，这证实了大型神经网络中观察到的参数量与鲁棒性之间的正相关性。我们还利用该模型阐明了现代Hopfield网络原型相具有对抗鲁棒性的原因。