Chenxue Yang and Mao Ye and Zijian Liu and Tao Li and Jiao Bao
Convergence Analysis of an Alternative Gradient Algorithm for NonNegative Matrix Factorization
201 - 210
2014
8
1
International Journal of Mathematical and Computational Sciences
https://publications.waset.org/pdf/9997720
https://publications.waset.org/vol/85
World Academy of Science, Engineering and Technology
Nonnegative matrix factorization (NMF) is a useful computational method to find basis information of multivariate nonnegative data. A popular approach to solve the NMF problem is the multiplicative update (MU) algorithm. But, it has some defects. So the columnwisely alternating gradient (cAG) algorithm was proposed. In this paper, we analyze convergence of the cAG algorithm and show advantages over the MU algorithm. The stability of the equilibrium point is used to prove the convergence of the cAG algorithm. A classic model is used to obtain the equilibrium point and the invariant sets are constructed to guarantee the integrity of the stability. Finally, the convergence conditions of the cAG algorithm are obtained, which help reducing the evaluation time and is confirmed in the experiments. By using the same method, the MU algorithm has zero divisor and is convergent at zero has been verified. In addition, the convergence conditions of the MU algorithm at zero are similar to that of the cAG algorithm at nonzero. However, it is meaningless to discuss the convergence at zero, which is not always the result that we want for NMF. Thus, we theoretically illustrate the advantages of the cAG algorithm.
Open Science Index 85, 2014