New SVD based initialization strategy for non-negative matrix factorization

Abstract

There are two problems need to be dealt with for Non-negative Matrix Factorization (NMF): choose a suitable rank of the factorization and provide a good initialization method for NMF algorithms. This paper aims to solve these two problems using Singular Value Decomposition (SVD). At first we extract the number of main components as the rank, actually this method is inspired from Turk and Pentland (1991) [15,16]. Second, we use the singular value and its vectors to initialize NMF algorithms. We title this new method as SVD-NMF. Boutsidis and Gollopoulos (2008) [2] provided the method titled NNDSVD to enhance initialization of NMF algorithms. They extracted the positive section and respective singular triplet information of the unit matrices {C(j)}j=1k which obtained by singular vector pairs based on SVD. In this strategy, they use the triplet information of SVD twice with low computational cost. The differences between SVD–NMF and NNDSVD are the once utilization of SVD in former method and different approximations of initializations for NMF algorithms. We report numerical experiments on two face databases ORL, YALE (C.U.C. Laboratory [10]; U.C. Version [17]) and one object database COIL-20 with two versions which are from Columbia University Image Library (Nene et al., 1996) [13]. Results show that SVD–NMF has faster convergence and provides an approximation with smaller errors than that of obtained by NNDSVD and random initialization.