  We present a fast variational Bayesian algorithm for performing non-negative
matrix factorisation and tri-factorisation. We show that our approach achieves
faster convergence per iteration and timestep (wall-clock) than Gibbs sampling
and non-probabilistic approaches, and do not require additional samples to
estimate the posterior. We show that in particular for matrix tri-factorisation
convergence is difficult, but our variational Bayesian approach offers a fast
solution, allowing the tri-factorisation approach to be used more effectively.
