Erwin E, Obermayer K, Schulten K
Beckman Institute, University of Illinois, Urbana-Champaign 61801.
Biol Cybern. 1992;67(1):35-45. doi: 10.1007/BF00201800.
We investigate the effect of various types of neighborhood function on the convergence rates and the presence or absence of metastable stationary states of Kohonen's self-organizing feature map algorithm in one dimension. We demonstrate that the time necessary to form a topographic representation of the unit interval [0, 1] may vary over several orders of magnitude depending on the range and also the shape of the neighborhood function, by which the weight changes of the neurons in the neighborhood of the winning neuron are scaled. We will prove that for neighborhood functions which are convex on an interval given by the length of the Kohonen chain there exist no metastable states. For all other neighborhood functions, metastable states are present and may trap the algorithm during the learning process. For the widely-used Gaussian function there exists a threshold for the width above which metastable states cannot exist. Due to the presence or absence of metastable states, convergence time is very sensitive to slight changes in the shape of the neighborhood function. Fastest convergence is achieved using neighborhood functions which are "convex" over a large range around the winner neuron and yet have large differences in value at neighboring neurons.
我们研究了各种类型的邻域函数对一维Kohonen自组织特征映射算法收敛速度以及亚稳态平稳状态存在与否的影响。我们证明，根据邻域函数的范围和形状，形成单位区间[0, 1]的地形表示所需的时间可能会在几个数量级上变化，通过该邻域函数可以缩放获胜神经元邻域内神经元的权重变化。我们将证明，对于在由Kohonen链长度给定的区间上凸的邻域函数，不存在亚稳态。对于所有其他邻域函数，存在亚稳态，并且在学习过程中可能会使算法陷入停滞。对于广泛使用的高斯函数，存在一个宽度阈值，高于该阈值不存在亚稳态。由于亚稳态的存在与否，收敛时间对邻域函数形状的微小变化非常敏感。使用在获胜神经元周围大范围“凸”但相邻神经元的值有很大差异的邻域函数可实现最快收敛。
