论文标题
深玻尔兹曼机器：任意深度的严格结果
Deep Boltzmann machines: rigorous results at arbitrary depth
论文作者
论文摘要
在具有高斯噪声和独立条目的淬灭系统的简化框架中，考虑了一类深玻尔兹曼机器。研究了$ k $层的旋转玻璃模型的淬火压力，仅在连续层之间进行相互作用。根据$ k $ Sherrington-Kirkpatrick型号的凸组合，发现压力的下限，并用于研究系统的退火和复制品对称式。通过Heilmann和Lieb引入的方法，确定了具有一维单体二聚体系统的地图，并用于严格控制在任意深度$ k $处的退火区域。该高噪声区域的压缩显示了处理层定位的显着现象。此外，在参数的合适区域中获得了模型的限制压力的复制对称下限，并且证明复制品对称压力具有独特的固定点。
A class of deep Boltzmann machines is considered in the simplified framework of a quenched system with Gaussian noise and independent entries. The quenched pressure of a $K$-layers spin glass model is studied allowing interactions only among consecutive layers. A lower bound for the pressure is found in terms of a convex combination of $K$ Sherrington-Kirkpatrick models and used to study the annealed and replica symmetric regimes of the system. A map with a one dimensional monomer-dimer system is identified and used to rigorously control the annealed region at arbitrary depth $K$ with the methods introduced by Heilmann and Lieb. The compression of this high noise region displays a remarkable phenomenon of localisation of the processing layers. Furthermore a replica symmetric lower bound for the limiting quenched pressure of the model is obtained in a suitable region of the parameters and the replica symmetric pressure is proved to have a unique stationary point.
