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.

Dense Hopfield网络在特征到原型的转换和对抗性鲁棒性方面具有特色。本研究通过研究无监督学习问题中教师-学生设置的p-body Hopfield网络的相图，在模式检索方面揭示了类似于原型学习和特征学习区域的铁磁相。我们发现在Nishimori线上，需要训练集的临界大小才能实现高效的模式检索。有趣的是，我们发现教师-学生设置的顺磁到铁磁的转变与直接模型（即随机模式）的顺磁到自旋玻璃的转变相一致。在Nishimori线之外，我们研究了学习性能与推断温度和数据集噪声的关系。此外，我们证明学生的p比教师更大可以给学生以对噪声的广泛容忍性。然后，我们导出了一个在零温下衡量这种学生对抗性鲁棒性的闭合形式表达式，证实了大型神经网络中观察到的参数数量与鲁棒性之间的正相关性。我们还使用我们的模型阐明了现代Hopfield网络的原型相为何具有对抗性鲁棒性。

教师-学生环境中的密集式霍普菲尔德网络