Mulier F, Cherkassky V
Department of Electrical Engineering, University of Minnesota, Minneapolis 55455, USA.
Neural Comput. 1995 Nov;7(6):1165-77. doi: 10.1162/neco.1995.7.6.1165.
Kohonen's self-organizing map, when described in a batch processing mode, can be interpreted as a statistical kernel smoothing problem. The batch SOM algorithm consists of two steps. First, the training data are partitioned according to the Voronoi regions of the map unit locations. Second, the units are updated by taking weighted centroids of the data falling into the Voronoi regions, with the weighing function given by the neighborhood. Then, the neighborhood width is decreased and steps 1, 2 are repeated. The second step can be interpreted as a statistical kernel smoothing problem where the neighborhood function corresponds to the kernel and neighborhood width corresponds to kernel span. To determine the new unit locations, kernel smoothing is applied to the centroids of the Voronoi regions in the topological space. This interpretation leads to some new insights concerning the role of the neighborhood and dimensionality reduction. It also strengthens the algorithm's connection with the Principal Curve algorithm. A generalized self-organizing algorithm is proposed, where the kernel smoothing step is replaced with an arbitrary nonparametric regression method.
当以批处理模式描述时，科霍宁自组织映射可以被解释为一个统计核平滑问题。批处理自组织映射算法由两个步骤组成。首先，根据映射单元位置的沃罗诺伊区域对训练数据进行划分。其次，通过获取落入沃罗诺伊区域的数据的加权质心来更新单元，其加权函数由邻域给出。然后，减小邻域宽度并重复步骤1和2。第二步可以被解释为一个统计核平滑问题，其中邻域函数对应于核，邻域宽度对应于核跨度。为了确定新的单元位置，在拓扑空间中对沃罗诺伊区域的质心应用核平滑。这种解释带来了一些关于邻域作用和降维的新见解。它还加强了该算法与主曲线算法的联系。提出了一种广义自组织算法，其中用任意非参数回归方法代替核平滑步骤。
