A Geometric Structure Preserving Non-negative Matrix Factorization for Data Representation

As a linear dimensionality reduction technique, non-negative matrix factorization (NMF) has been widely used in many fields. However, NMF can only perform semantic factorization in Euclidean space, and it fails to discover the intrinsic geometrical structure of high-dimensional data distribution. To address this issue, in this paper, we propose a new non-negative matrix factorization algorithm, known as the structure preserving non-negative matrix factorization (SPNMF). Compared with the existing NMF, our SPNMF method effectively exploits the local affinity structure and distant repulsion structure among data samples. Specifically, we incorporate the local and distant structure preservation terms into the NMF framework and then give an alternative optimization method for SPNMF. Due to prior knowledge from the structure preservation term, SPNMF can learn a good low-dimensional representation. Experimental results on some facial image dataset clustering show the significantly improved performance of SPNMF compared with other state-of-the-art algorithms.