Abstract

Algorithms for decompositions of matrices are of central importance in machine learning, signal processing and information retrieval, with SVD and NMF (Nonnegative Matrix Factorisation) being the most widely used examples. Probabilistic interpretations of matrix factorisation models are also well known and are useful in many applications (Salakhutdinov and Mnih 2008; Cemgil 2009; Fevotte et. al. 2009). In the recent years, decompositions of multiway arrays, known as tensor factorisations have gained significant popularity for the analysis of large data sets with more than two entities (Kolda and Bader, 2009; Cichocki et. al. 2008; Mohamed 2011). We will discuss a subset of these models from a statistical modelling perspective, building upon probabilistic generative models and generalised linear models (McCulloch and Nelder). In both views, the factorisation is implicit in a well-defined statistical model and factorisations can be computed via maximum likelihood.

We express a tensor factorisation model using a factor graph and the factor tensors are optimised iteratively. In each iteration, the update equation can be implemented by a message passing algorithm, reminiscent to variable elimination in a discrete graphical model. This setting provides a structured and efficient approach that enables very easy development of application specific custom models, as well as algorithms for the so called coupled (collective) factorisations where an arbitrary set of tensors are factorised simultaneously with shared factors. Extensions to full Bayesian inference for model selection, via variational approximations or MCMC are also feasible. Well known models of multiway analysis such as Nonnegative Matrix Factorisation (NMF), Parafac, Tucker, and audio processing (Convolutive NMF , NMF2D, SF-SSNTF) appear as special cases and new extensions can easily be developed. We will illustrate the approach with applications in audio and music processing and link prediction for recommendation.