On the convergence of multiplicative update algorithms for nonnegative matrix factorization

Nonnegative matrix factorization (NMF) is useful to find basis information of nonnegative data. Currently, multiplicative updates are a simple and popular way to find the factorization. However, for the common NMF approach of minimizing the Euclidean distance between approximate and true values, no proof has shown that multiplicative updates converge to a stationary point of the NMF optimization problem. Stationarity is important as it is a necessary condition of a local minimum. This paper discusses the difficulty of proving the convergence. We propose slight modifications of existing updates and prove their convergence. Techniques invented in this paper may be applied to prove the convergence for other bound-constrained optimization problems.