Statistics and Computing

, Volume 22, Issue 4, pp 945–957

A hierarchical model for ordinal matrix factorization


DOI: 10.1007/s11222-011-9264-x

Cite this article as:
Paquet, U., Thomson, B. & Winther, O. Stat Comput (2012) 22: 945. doi:10.1007/s11222-011-9264-x


This paper proposes a hierarchical probabilistic model for ordinal matrix factorization. Unlike previous approaches, we model the ordinal nature of the data and take a principled approach to incorporating priors for the hidden variables. Two algorithms are presented for inference, one based on Gibbs sampling and one based on variational Bayes. Importantly, these algorithms may be implemented in the factorization of very large matrices with missing entries.

The model is evaluated on a collaborative filtering task, where users have rated a collection of movies and the system is asked to predict their ratings for other movies. The Netflix data set is used for evaluation, which consists of around 100 million ratings. Using root mean-squared error (RMSE) as an evaluation metric, results show that the suggested model outperforms alternative factorization techniques. Results also show how Gibbs sampling outperforms variational Bayes on this task, despite the large number of ratings and model parameters. Matlab implementations of the proposed algorithms are available from


Large scale machine learningCollaborative filteringOrdinal regressionLow rank matrix decompositionHierarchial modellingBayesian inferenceVariational BayesGibbs sampling

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Microsoft Research CambridgeCambridgeUK
  2. 2.Engineering DepartmentUniversity of CambridgeCambridgeUK
  3. 3.Informatics and Mathematical ModellingTechnical University of DenmarkLyngbyDenmark