A Bayesian approach to the Lee–Seung update rules for NMF

• A new variational Bayes algorithm for non-negative matrix factorization is proposed.
• The algorithm VBNMF generalizes the classical Lee–Seung multiplicative update rules.
• The Lee–Seung rules are obtained from a MAP approximation of the VBNMF algorithm.
• The VBNMF can provide model order selection and automatic relevance detection.

NMF is a Blind Source Separation technique decomposing multivariate non-negative data sets into meaningful non-negative basis components and non-negative weights. In its canonical form an NMF algorithm was proposed by Lee and Seung (1999) [31] employing multiplicative update rules. In this study we show how the latter follow from a new variational Bayes NMF algorithm VBNMF employing a Gaussian noise kernel.