Proximal Gradient Method For Convex And Semi-Nonnegative Matrix Factorization

Nonnegative matrix factorization(NMF) is an important area of high dimension reduction with non-negative data and receive considerable attentions. Convex and seminonnegative matrix factorization(Convex-NMF and Semi-NMF) are introduced by Ding et al[9]for more general data, for example, mixed sign data, which can capture the key feature of the data. Ding et al[9]propose multiplication update strategy for Convex-NMF and Semi-NMF, while their proposed algorithms converge generally very slow for large dimensional data. In this thesis we apply the proximal gradient method for Convex-NMF and Semi-NMF. Using nonsmooth analysis and Kurdyka-?ojasiewicz fuction property,we prove the convergence theory of the proposed algorithms. In recent years, Nesterovtype extrapolation techniques are often used to accelerate the proximal gradient method for convex problem. Therefore we use one of these techniques to accelerate the proximal gradient method for Convex-NMF and Semi-NMF. And numerical examples show that the accelerate versions are fast and the convergence speeds have been greatly improved.We test the algorithms on both synthetic and real-world datasets and compared them with the previous Ding et al’s algorithms. The numerical results show that our algorithms are faster 10 times than Ding et al’s algorithms in CPU times and iteration steps. In terms of sparsity and orthogonality, our algorithms have a slightly advantages than Ding et al’s algorithms. Extensive experiments on synthetic data for clustering suggest that our methods are more reliable.