Statistics and Computing

, Volume 22, Issue 4, pp 945-957

First online:

A hierarchical model for ordinal matrix factorization

  • Ulrich PaquetAffiliated withMicrosoft Research Cambridge Email author 
  • , Blaise ThomsonAffiliated withEngineering Department, University of Cambridge
  • , Ole WintherAffiliated withInformatics and Mathematical Modelling, Technical University of Denmark

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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 cogsys.​imm.​dtu.​dk/​ordinalmatrixfac​torization.


Large scale machine learning Collaborative filtering Ordinal regression Low rank matrix decomposition Hierarchial modelling Bayesian inference Variational Bayes Gibbs sampling