Bayesian matrix completion: prior specification
This work addresses the problem of efficient Bayesian inference in matrix completion for applications like recommender systems, but it is incremental as it builds on existing priors and methods.
The paper reviews and introduces conjugate priors for Bayesian matrix completion, showing that gamma and discrete priors for singular values enable efficient Gibbs sampling or Variational Bayes, with MAP estimates linked to nuclear norm minimization, and validates these on simulated and real datasets like MovieLens and Netflix.
Low-rank matrix estimation from incomplete measurements recently received increased attention due to the emergence of several challenging applications, such as recommender systems; see in particular the famous Netflix challenge. While the behaviour of algorithms based on nuclear norm minimization is now well understood, an as yet unexplored avenue of research is the behaviour of Bayesian algorithms in this context. In this paper, we briefly review the priors used in the Bayesian literature for matrix completion. A standard approach is to assign an inverse gamma prior to the singular values of a certain singular value decomposition of the matrix of interest; this prior is conjugate. However, we show that two other types of priors (again for the singular values) may be conjugate for this model: a gamma prior, and a discrete prior. Conjugacy is very convenient, as it makes it possible to implement either Gibbs sampling or Variational Bayes. Interestingly enough, the maximum a posteriori for these different priors is related to the nuclear norm minimization problems. We also compare all these priors on simulated datasets, and on the classical MovieLens and Netflix datasets.