  We introduce a new non-negative matrix factorization (NMF) method for ordinal
data, called OrdNMF. Ordinal data are categorical data which exhibit a natural
ordering between the categories. In particular, they can be found in
recommender systems, either with explicit data (such as ratings) or implicit
data (such as quantized play counts). OrdNMF is a probabilistic latent factor
model that generalizes Bernoulli-Poisson factorization (BePoF) and Poisson
factorization (PF) applied to binarized data. Contrary to these methods, OrdNMF
circumvents binarization and can exploit a more informative representation of
the data. We design an efficient variational algorithm based on a suitable
model augmentation and related to variational PF. In particular, our algorithm
preserves the scalability of PF and can be applied to huge sparse datasets. We
report recommendation experiments on explicit and implicit datasets, and show
that OrdNMF outperforms BePoF and PF applied to binarized data.
