We consider restricted Boltzmann machines with a binary visible layer and a Gaussian hidden layer trained by an unlabelled dataset composed of noisy realizations of a single ground pattern. We develop a statistical mechanics framework to describe the network generative capabilities, by exploiting the replica trick and assuming self-averaging of the underlying order parameters (i.e., replica symmetry). In particular, we outline the effective control parameters (e.g., the relative number of weights to be trained, the regularization parameter), whose tuning can yield qualitatively-different operative regimes. Further, we provide analytical and numerical evidence for the existence of a sub-region in the space of the hyperparameters where replica-symmetry breaking occurs.

翻译：我们研究具有二元可见层和高斯隐藏层的受限玻尔兹曼机，其训练数据由单个基础模式的噪声实现组成。通过利用副本技巧并假设底层序参量具有自平均性（即副本对称性），我们建立了一个统计力学框架来描述网络的生成能力。特别地，我们阐明了有效控制参数（例如待训练权重的相对数量、正则化参数）的调节可导致性质不同的操作模式。此外，我们提供了理论和数值证据，证明在超参数空间存在一个子区域会发生副本对称性破缺。