Abstract: Aiming at the problem of the error existed from the hidden layer to the output layer of Extreme Learning Machine(ELM), it was found the analysis revealed that the error came from the process of solving the Moore-Penrose generalized inverse matrix H^† of the hidden layer output matrix H,that revaled the matrix H^† H was deviated from the identity matrix. The appropriate output matrix H was able to be selected according to the degree of deviation to obtain a smaller training error. According to the definitions of the generalized inverse matrix and auxiliary matrix,the target matrix H^† H and the error index L21-norm were firstly determined. Then,the experimental analysis showed that the L21-norm of H^† H was linearly related to the ELM error. Finally,Gaussian filtering was introduced to reduce the noise of the target matrix,which effectively reduced the L21-norm of the target matrix and the ELM error,achieving the purpose of optimizing the ELM algorithm.

Key words: Extreme Learning Machine (ELM), Moore-Penrose generalized inverse matrix, L21-norm, linear correlation, Gaussian filtering

摘要： 针对极限学习机（ELM）中隐藏层到输出层存在误差的问题，通过分析发现误差来源于求解隐藏层输出矩阵H的Moore-Penrose广义逆矩阵Η^†的过程，即矩阵H^†H与单位矩阵有偏差，可根据偏差的程度来选择合适的输出矩阵H以获得较小的训练误差。根据广义逆矩阵和辅助矩阵的定义，首先确定了目标矩阵H^†H和误差指标L21范数，其次通过实验分析表明H^†H的L21范数与ELM的误差呈显著线性相关，最后通过引入Gaussian滤波对目标矩阵进行降噪处理，由此有效降低了目标矩阵的L21范数，同时降低了ELM的误差，达到优化ELM算法的目的。

关键词: 极限学习机, Moore-Penrose广义逆矩阵, L21范数, 线性相关, Gaussian滤波