# Bayesian matrix completion: prior specification

arXiv: https://arxiv.org/abs/1406.1440 · HTML: https://mlsanity.com/paper/1406.1440/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jun 2014.

## Built on by (most cited first, 3 of 3)

- [Concentration properties of fractional posterior in 1-bit matrix completion](https://mlsanity.com/paper/2404.08969/) (2024)
- [Simulation comparisons between Bayesian and de-biased estimators in low-rank matrix completion](https://mlsanity.com/paper/2103.11749/) (2021)
- [PAC-Bayesian Matrix Completion with a Spectral Scaled Student Prior](https://mlsanity.com/paper/2104.08191/) (2021)

## Abstract (arXiv)

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.