基于凸函数理论的Hopfield网络稳定性分析

讨论连接权值不对称或激活函数非单调的离散时间Hopfield网络稳定性分析。引入新的能量函数，利用凸函数的性质证明随状态的更新网络能量函数单调下降从而得出网络收敛的充分条件。对于激活函数为非单调的连续函数而网络连接权值对称，则当网络连接权值矩阵的最大特征值和神经元激活函数的导数下确界之积大于-1时，网络全并行收敛。对于网络激活函数为单调连续函数，网络连接权值为非对称矩阵时，神经元激活函数导数的最大值和连接权值矩阵的2-范数之积小于1时，网络全并行收敛。

Stability Analysis to Hopfield Network Based on Convex Function Theory

This paper discusses stability of discrete time Hopfield network with the sequential weight which is asymmetry or with the activation function which is non-monotonic. New energy function is imported, then it proved that the energy function of network is declined monotonously with the updating of the state using the properties of convex function, gained the sufficient condition of the network convergence in the end. To the activation function that is non-monotonic sequential function, when product of the maximum characteristic value of the network connecting weight matrix and the infimum of the derivative to the activation function of the nerve cell more than -1, the network convergences concurrently. At the same time, to the network activation function is monotonous sequential function, and the network connecting weight is asymmetry matrix~ when the product of the maximum value of derivative to the nerve cell and the matrix of the two-norm to the connecting weight value is less than 1, the network convergences concurrently.