Algorithms for Non-negative Matrix Factorization

Nonnegative matrix factorization (NMF) has previously been shown to be a useful decomposition for multivariate data. Two different multiplicative algorithmsfor NMF are analyzed. They differ only slightly in the multiplicative factor used in the update rules. One algorithm can be shown to minimize the conventional least squares error while the other minimizes the generalized Kullback-Leibler divergence. The monotonic convergence of both algorithms can be proven using an auxiliary function analogousto that used for proving convergence of the Expectation Maximization algorithm. The algorithms can also be interpreted as diagonally rescaledgradient descent, where the rescaling factor is optimally chosen to ensure convergence.