We define a message-passing algorithm for computing magnetizations in Restricted Boltzmann machines, which are Ising models on bipartite graphs introduced as neural network models for probability distributions over spin configurations. To model nontrivial statistical dependencies between the spins' couplings, we assume that the rectangular coupling matrix is drawn from an arbitrary bi-rotation invariant random matrix ensemble. Using the dynamical functional method of statistical mechanics we exactly analyze the dynamics of the algorithm in the large system limit. We prove the global convergence of the algorithm under a stability criterion and compute asymptotic convergence rates showing excellent agreement with numerical simulations.

中文翻译：

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限制玻尔兹曼机学习的动态平均场理论

我们定义了一种消息传递算法，用于计算受限玻尔兹曼机中的磁化强度，这是二部图上的 Ising 模型，作为自旋配置概率分布的神经网络模型引入。为了模拟自旋耦合之间的非平凡统计依赖关系，我们假设矩形耦合矩阵是从任意双旋转不变随机矩阵系综中绘制的。运用统计力学的动力学泛函方法，准确分析了算法在大系统极限下的动力学。我们证明了算法在稳定性标准下的全局收敛性，并计算了渐近收敛率，表明与数值模拟非常吻合。