Chinese Journal of Computers   Full Text
Title	Bilinear Form-Based Non-Negative Matrix Set Factorization
Authors	LI Le ZHANG Yu-Jin
Address	(Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, Beijing 100084) (Department of Electronic Engineering, Tsinghua University, Beijing 100084)
Year	2009
Issue	No.8(1536—1549)
Abstract &   Background	Abstract Non-negative Matrix Factorization (NMF) is a popular technique for representations of non-negative multivariate data. While treating a set of matrices, NMF is confronted with two main problems (unsatisfactory accuracy of representation and bad generality). In this paper, Non-negative Matrix Set Factorization (NMSF) is conceived to overcome the two problems and to retain NMF’s good properties. Under the frame of NMSF, Bilinear Form-Based Non-negative Matrix Set Factorization (BFBNMSF) is systematically studied, and a monotonic algorithm of BFBNMSF is put forward. Theoretical analysis and experimental results show that while processing a data matrix-set, BFBNMSF results in more accurate representation and holds better generality than NMF, therefore it tends to extract more essential features of data matrix sets than NMF. Keywords Non-negative Matrix Set Factorization (NMSF); bilinear form; Nonnegative Matrix Factorization (NMF); multivariate data representation; image representation; feature extraction Background Non-negative matrix factorization (NMF) is a relatively new but more and more popular method for representations of nonnegative multivariate data. NMF can reveal the latent structure, feature or pattern in the data, so that it has been applied in several research fields. However, NMF is confronted with two main problems (unsatisfactory accuracy of representation and bad generality) while the processed is a matrix set, because the object processed by NMF is intrinsically a set of vectors, and because the necessary vectorization for every matrix in the processed matrix set often make the dimension of the parameter in NMF learning to be much greater than training sample size so that the learning becomes typical small-sample learning. In this paper supported by Project 60872084 of National Natural Science Foundation of China, non-negative matrix set factorization (NMSF) is conceived to overcome the problems above and to retain NMF’s good properties. As opposed to NMF, NMSF directly processes original data matrices rather than vectorization results of them. Under the frame of NMSF, bilinear form-based non-negative matrix set factorization (BFBNMSF) is systematically studied, and a monotonic algorithm of BFBNMSF is put forward. Theoretical analysis and experimental results show that while processing a data matrix set, BFBNMSF results in more accurate representation and holds better generality than NMF, therefore it tends to extract more essential features of data matrix sets than NMF.