伊辛模型的非回溯算子及其在吸引子神经网络中的应用
Non-backtracking operator for Ising model and its application in attractor neural networks
摘要：
非回溯算子最近被证明是稀疏图中谱聚类的一种救赎。本文考虑具有一般耦合分布的一般图上的伊辛模型的非回溯算子，在顺磁固定点上将置信传播算法线性化.研究了该算子的谱，解析计算了体的锐边和体外可能的实本征值随耦合和温度的变化。我们展示了该算子在吸引子神经网络中的应用。在热力学极限下，我们的结果恢复了Hopfield模型的复型方法得到的相边界。在Hopfield模型的单个实例中，其特征向量可以同时用于检索所有模式。我们还给出了一个例子，如何控制神经网络，即使网络更稀疏，同时保持模式稳定，使用非回溯算子和矩阵扰动理论。
英文摘要：
The non-backtracking operator was recently shown to give a redemption for spectral clustering in sparse graphs. In this paper we consider non-backtracking operator for Ising model on a general graph with a general coupling distribution by linearizing Belief Propagation algorithm at paramagnetic fixed-point. The spectrum of the operator is studied, the sharp edge of bulk and possible real eigenvalues outside the bulk are computed analytically as a function of couplings and temperature. We show the applications of the operator in attractor neural networks. At thermodynamic limit, our result recovers the phase boundaries of Hopfield model obtained by replica method. On single instances of Hopfield model, its eigenvectors can be used to retrieve all patterns simultaneously. We also give an example on how to control the neural networks, i.e. making network more sparse while keeping patterns stable, using the non-backtracking operator and matrix perturbation theory.
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