# Ordinal Non-negative Matrix Factorization for Recommendation

arXiv: https://arxiv.org/abs/2006.01034 · HTML: https://mlsanity.com/paper/2006.01034/

## Key facts

- Cited by 8 papers in the map (data through 7 Oct 2026).
- First posted on arXiv 1 Jun 2020.

## Built on by (most cited first, 8 of 8)

- [Recommender systems based on graph embedding techniques: A comprehensive review](https://mlsanity.com/paper/2109.09587/) (2021)
- [Joint Majorization-Minimization for Nonnegative Matrix Factorization with the β-divergence](https://mlsanity.com/paper/2106.15214/) (2021)
- [Majorization-minimization for Sparse Nonnegative Matrix Factorization with the β-divergence](https://mlsanity.com/paper/2207.06316/) (2022)
- [Towards Ordinal Data Science](https://mlsanity.com/paper/2307.09477/) (2023)
- [Modeling Human Responses by Ordinal Archetypal Analysis](https://mlsanity.com/paper/2409.07934/) (2024)
- [Grouping Search Results with Product Graphs in E-commerce Platforms](https://mlsanity.com/paper/2109.09349/) (2021)
- [Ordinal Graph Gamma Belief Network for Social Recommender Systems](https://mlsanity.com/paper/2209.05106/) (2022)
- [Robust Capped lp-Norm Support Vector Ordinal Regression](https://mlsanity.com/paper/2404.16616/) (2024)

## Abstract (arXiv)

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.