Multiple Graph Regularized Non-negative Matrix Factorization

It is an era of information explosion now, people need to face vast amounts of datum and high-dimensional data. Therefore, how to represent data properly is particularly important and there is also research significance on the low-dimensional data representation. Matrix factorization technique is one of the important methods to realize data dimension reduction,and the researchers have developed many classic matrix factorization algorithms. In practice,it is found that matrix factorization algorithm can realize the data dimension reduction, but it allows that the original data matrix and the result of factorization is negative. It is not conform to the physical meaning of data. Lee proposed non-negative matrix factorization(NMF)algorithm, which require all elements of matrix is non-negative. It makes the decomposition on the form and the decomposition results can be interpreted. In many tasks of pattern recognition, computer vision and image clustering, NMF has become an effective method in recent years for data representation. Therefore many algorithms based on NMF proposed one after other.The convex non-negative matrix factorization(CNMF) is a variation of non-negative matrix factorization(NMF) in which each cluster is expressed by a linear combination of the data points and each data point is represented by a linear combination of the cluster centers.When there exists nonlinearity in the manifold structure, both NMF and CNMF are incapable of characterizing the geometric structure of the data. Recently, manifold learning theory has received the widespread attention in the field of machine learning, it has become a hot topic to combine the manifold learning ideas with NMF. Among them, Graph regularized non-negative matrix factorization(GNMF) has received considerable attention due to uncovering the intrinsic geometric structure of the data space, but it can not estimate the intrinsic manifolds in a principled way. In this article, we propose a novel algorithm, called Multiple Graph Regularized Non-negative matrix factorization(MGNMF), which maximally approximate the intrinsic manifolds of the sample space. The clustering experiments on the PIE and ORL image databases demonstrate that MGNMF is superior to other approaches. It has a higher accuracy.