基于Huber损失的非负矩阵分解算法

Huber Loss Based Nonnegative Matrix Factorization Algorithm

非负矩阵分解(Nonnegative Matrix Factorization)算法能为原始数据找到非负的、线性的矩阵表示且保留了数据的本质特征,已被成功应用于多个领域。经典的NMF算法及其变体算法大部分使用均方误差函数来度量重建误差,在许多任务中已经显示出其有效性,但它在处理含有噪声的数据时仍然面临一些困难。Huber损失函数对较小的残差执行的惩罚与均方误差损失函数相同,对较大的残差执行的惩罚是线性增长的,因此与均方误差损失函数相比,Huber损失函数具有更强的鲁棒性;已有研究证明L_(2,1)范数稀疏正则项在机器学习的分类和聚类模型中具有特征选择作用。结合两者的优点,文中提出了一种基于Huber损失函数且融入L_(2,1)范数正则项的非负矩阵分解聚类模型,并给出了基于投影梯度更新规则的优化过程。在多组数据集上将所提算法与经典的多种聚类算法进行对比,实验结果验证了所提算法的有效性。

Non-negative matrix factorization(NMF)algorithm can find a non-negative and linear matrix representation and retains the essential characteristics of the original data,it has been successfully applied to many fields.The classical NMF algorithm and its variant algorithms mostly use the mean square error function to measure the reconstruction error,which has been shown to be effective in many tasks,but it still faces some difficulties in dealing with noise-containing data.The Huber loss function performs the same penalty for the smaller residual as the mean square error loss function,and the penalty for the larger residual is linearly grown,so the Huber loss function is more robust than the mean square error loss function.It has been proved that the L 2,1 norm sparse regularization term is a feature selection function in the classification and clustering model of machine learning.Therefore,combining the advantages of the two,a non-negative matrix factorization clustering model based on Huber loss function and incorporating L 2,1 norm regularization term is proposed,and an effective optimization procedure based on projected gradient method to update variables is given.Compared with the classical NMF multi-clustering algorithm on multiple sets of datasets,the experimental results show the effectiveness of the proposed algorithm.