A convergent algorithm for bi-orthogonal nonnegative matrix tri-factorization

A convergent algorithm for nonnegative matrix factorization (NMF) with orthogonality constraints imposed on both basis and coefficient matrices is proposed in this paper. This factorization concept was first introduced by Ding et al. (Proceedings of 12th ACM SIGKDD international conference on knowledge discovery and data mining, pp 126–135, 2006) with intent to further improve clustering capability of NMF. However, as the original algorithm was developed based on multiplicative update rules, the convergence of the algorithm cannot be guaranteed. In this paper, we utilize the technique presented in our previous work Mirzal (J Comput Appl Math 260:149–166, 2014a; Proceedings of the first international conference on advanced data and information engineering (DaEng-2013). Springer, pp 177–184, 2014b; IEEE/ACM Trans Comput Biol Bioinform 11(6):1208–1217, 2014c) to develop a convergent algorithm for this problem and prove that it converges to a stationary point inside the solution space. As it is very hard to numerically show the convergence of an NMF algorithm due to the slow convergence and numerical precision issues, experiments are instead performed to evaluate whether the algorithm has the nonincreasing property (a necessary condition for the convergence) where it is shown that the algorithm has this property. Further, clustering capability of the algorithm is also inspected by using Reuters-21578 data corpus.