Dipartimento di Matematica e Fisica Ennio De Giorgi, Università del Salento, 73100 Lecce, Italy.
Institut für Mathematik, Universität Zürich, CH-8057 Zürich, Switzerland.
Phys Rev E. 2018 Feb;97(2-1):022310. doi: 10.1103/PhysRevE.97.022310.
Restricted Boltzmann machines are described by the Gibbs measure of a bipartite spin glass, which in turn can be seen as a generalized Hopfield network. This equivalence allows us to characterize the state of these systems in terms of their retrieval capabilities, both at low and high load, of pure states. We study the paramagnetic-spin glass and the spin glass-retrieval phase transitions, as the pattern (i.e., weight) distribution and spin (i.e., unit) priors vary smoothly from Gaussian real variables to Boolean discrete variables. Our analysis shows that the presence of a retrieval phase is robust and not peculiar to the standard Hopfield model with Boolean patterns. The retrieval region becomes larger when the pattern entries and retrieval units get more peaked and, conversely, when the hidden units acquire a broader prior and therefore have a stronger response to high fields. Moreover, at low load retrieval always exists below some critical temperature, for every pattern distribution ranging from the Boolean to the Gaussian case.
受限玻尔兹曼机由二部自旋玻璃的吉布斯测度描述，而自旋玻璃又可以看作广义的霍普菲尔德网络。这种等价关系使我们能够根据其检索能力来描述这些系统的状态，无论是在低负载还是高负载下，都可以对纯态进行检索。我们研究了顺磁-自旋玻璃和自旋玻璃-检索相变，因为模式（即权重）分布和自旋（即单元）先验从高斯实变量到布尔离散变量平滑地变化。我们的分析表明，检索相的存在是稳健的，并且不仅限于具有布尔模式的标准霍普菲尔德模型。当模式条目和检索单元变得更加尖锐时，检索区域会变大，而当隐藏单元获得更广泛的先验时，它们对高场的响应会更强。此外，在低负载下，对于从布尔到高斯的每种模式分布，在某个临界温度以下总是存在检索。
