基于线性投影结构的非负矩阵分解

Linear Projection-based Non-negative Matrix Factorization

非负矩阵分解(Non-negative matrix factorization,NMF)是一个近年来非常流行的非负数据处理方法,它常用于维数约减、特征提取和数据挖掘等.NMF定义中采用的数学模型基于非线性投影结构构造,这决定了NMF降维需借助计算量很大的迭代操作来实现.此外,由此模型提取的NMF特征常不稀疏,这与NMF的设计期望相差甚远.为一并解决上述两个问题,本文提出了一个新的模型—基于线性投影结构的NMF(Linear projection-based NMF,LPBNMF),并构造了一个单调的LPBNMF算法.从数学的角度看,LPBNMF可理解为实现NMF的一种特殊方式.LPBNMF降维通过线性变换来完成,它所采用的数学模型的自身结构特点决定了由其得到的特征一定非常稀疏.大量的比较实验表明,LPBNMF的降维效率显著高于NMF,LPBNMF特征明显比NMF特征更稀疏和局部化.最后,基于AR人脸数据库的实验揭示,LPBNMF特征比NMF、LDA以及PCA等特征更适合于用最近邻分类法处理有遮挡人脸识别问题.

Non-negative matrix factorization （NMF） is a newly popular method for non-negative dimensionality reduction, feature extraction, data mining, etc. The mathematical model in NMF definition is based on nonlinear projection, therefore dimension reduction by NMF is implemented by iterative updates which lead to high computational load. Additionally, NMF features extracted by this model are usually not very sparse, and this fails to meet the expectation of designing NMF. To simultaneously resolve the above two problems, this paper proposes a new model, linear projection-based NMF （LPBNMF）, and designs an monotonic algorithm for it. From mathematical point of view, LPBNMF is a special mode for implementing NMF, which linearly implements dimension reduction. The high sparseness of LPBNMF features is assured by the inherent characteristics of its mathematic model. The comparison experiments validate that dimension reduction by LPBNMF is much more efficient than that by NMF, and that LPBNMF features are much more sparse and localized than NMF ones. Finally, experiments based on AR face database indicate that LPBNMF features are more suitable for nearest neighbor classification-based occluded face recognition than NMF, LDA, and PCA ones.