We consider a special type of restricted Boltzmann machine (RBM), namely a Gaussian-spherical RBM where the visible units have Gaussian priors while the vector of hidden variables is constrained to stay on an ##IMG## [http://ej.iop.org/images/1751-8121/53/18/184002/aab79f3ieqn1.gif] {${\mathbb{L}}_{2}$} sphere. The spherical constraint having the advantage to admit exact asymptotic treatments, various scaling regimes are explicitly identified based solely on the spectral properties of the coupling matrix (also called weight matrix of the RBM). Incidentally these happen to be formally related to similar scaling behaviors obtained in a different context dealing with spatial condensation of zero range processes. More specifically, when the spectrum of the coupling matrix is doubly degenerated an exact treatment can be proposed to deal with finite size effects. Interestingly the known parallel between the ferromagnetic transition of the spherical model and the ...

中文翻译：

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高斯球形受限玻尔兹曼机

我们考虑一种特殊类型的受限玻尔兹曼机（RBM），即高斯球面RBM，其中可见单位具有高斯先验，而隐藏变量的矢量被约束在## IMG ##上。 iop.org/images/1751-8121/53/18/184002/aab79f3ieqn1.gif] {$ {\ mathbb {L}} _ {2} $}球形。球形约束具有允许进行精确渐近处理的优势，仅基于耦合矩阵（也称为RBM的权重矩阵）的光谱特性，可以明确标识各种缩放方案。偶然地，这些恰好与在处理零范围过程的空间凝结的不同上下文中获得的类似缩放行为正式相关。进一步来说，当耦合矩阵的光谱被双重退化时，可以提出一种精确的处理方法来处理有限的尺寸效应。有趣的是，球形模型的铁磁跃迁与磁导率之间的已知平行关系...