发表机构
Dipartimento di Matematica, Sapienza Università di Roma; GNFM, Istituto Nazionale di Alta Matematica Francesco Severi (INdAM); CNR-Nanotec, Unità di Lecce（罗马大学数学系; 高等数学弗朗切斯科·塞韦里国家研究所; 莱切分部纳米技术研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究模块化受限玻尔兹曼机，利用Hopfield模型与RBM的对偶性，通过反Hebbian耦合实现模式解缠，并训练网络进行联合分类与解缠任务。
AI 中文摘要
我们考虑一个由 $L$ 个 Hopfield 模型（HMs）组成的模块化联想神经网络，这些模型通过耦合使得模块内相互作用是 Hebbian 的，而模块间相互作用是反 Hebbian 的；这种竞争性耦合已知能使网络具备模式解缠能力。该系统的积分表示与 $L$ 个受限玻尔兹曼机（RBMs）的集合一致，这些 RBM 的隐藏层相互耦合，从而将 HM-RBM 对偶性扩展到模块化设置。然后，我们通过一步对比散度训练这个模块化 RBM，以执行一项分类任务，其中编码在可见层上的查询被映射到从隐藏层读取的 $L$ 元组标签上。当查询由 $L$ 个平均相互正交的模式组成时，我们证明，根据 HM-RBM 等价性所建议的、作为训练数据集上的经验均值获得的 RBM 权重，构成学习动力学的一个固定点，并且我们推导出残差漂移的显式非渐近界。然后，我们转向一个更困难的分类任务，其中每个模块被查询一个由 $L$ 个模式组成的混合体，我们通过数值实验表明，RBM 权重的相同设置仍然提供有效的设置，使网络能够联合分类和解缠混合体。
英文摘要
We consider a modular associative neural network made of $L$ Hopfield models (HMs), coupled so that intra-module interactions are Hebbian and inter-module interactions are anti-Hebbian; this competitive coupling is known to endow the network with pattern-disentanglement capabilities. The integral representation of this system coincides with an assembly of $L$ restricted Boltzmann machines (RBMs) whose hidden layers are coupled, thereby extending the HM-RBM duality to the modular setting. We then train this modular RBM, via one-step contrastive divergence, to perform a classification task in which a query encoded on the visible layers is mapped onto an $L$-tuple of labels read off the hidden layers. When the query is composed of $L$ patterns that are mutually orthogonal on average, we prove that the RBM weights obtained as empirical means over the training dataset, as suggested by the HM-RBM equivalence, constitute a fixed point of the learning dynamics, and we derive an explicit, non-asymptotic bound on the residual drift. We then turn to a harder classification task in which each module is queried with a mixture of $L$ patterns and we show numerically that the same setting for the RBM weights still provides an effective set-up, letting the network jointly classify and disentangle the mixture.
