We introduce negative binomial matrix factorization (NBMF), a matrix factorization technique specially designed for analyzing over-dispersed count data. It can be viewed as an extension of Poisson factorization (PF) perturbed by a multiplicative term which models exposure. This term brings a degree of freedom for controlling the dispersion, making NBMF more robust to outliers. We describe a majorization-minimization (MM) algorithm for a maximum likelihood estimation of the parameters. We provide results on a recommendation task and demonstrate the ability of NBMF to efficiently exploit raw data.

We introduce negative binomial matrix factorization (NBMF), a matrix factorization technique specially designed for analyzing over-dispersed count data. It can be viewed as an extension of Poisson matrix factorization (PF) perturbed by a multiplicative term which models exposure. This term brings a degree of freedom for controlling the dispersion, making NBMF more robust to outliers. We show that NBMF allows to skip traditional pre-processing stages, such as binarization, which lead to loss of information. Two estimation approaches are presented: maximum likelihood and variational Bayes inference. We test our model with a recommendation task and show its ability to predict user tastes with better precision than PF.

Count data are often used in recommender systems: they are widespread (song play counts, product purchases, clicks on web pages) and can reveal user preference without any explicit rating from the user. Such data are known to be sparse, over-dispersed and bursty, which makes their direct use in recommender systems challenging, often leading to pre-processing steps such as binarization. The aim of this paper is to build recommender systems from these raw data, by means of the recently proposed compound Poisson Factorization (cPF). The paper contributions are three-fold: we present a unified framework for discrete data (dcPF), leading to an adaptive and scalable algorithm; we show that our framework achieves a trade-off between Poisson Factorization (PF) applied to raw and binarized data; we study four specific instances that are relevant to recommendation and exhibit new links with combinatorics. Experiments with three different datasets show that dcPF is able to effectively adjust to over-dispersion, leading to better recommendation scores when compared with PF on either raw or binarized data.

Non-negative matrix factorization (NMF) has become a well-established class of methods for the analysis of non-negative data. In particular, a lot of effort has been devoted to probabilistic NMF, namely estimation or inference tasks in probabilistic models describing the data, based for example on Poisson or exponential likelihoods. When dealing with time series data, several works have proposed to model the evolution of the activation coefficients as a non-negative Markov chain, most of the time in relation with the Gamma distribution, giving rise to so-called temporal NMF models. In this paper, we review four Gamma Markov chains of the NMF literature, and show that they all share the same drawback: the absence of a well-defined stationary distribution. We then introduce a fifth process, an overlooked model of the time series literature named BGAR(1), which overcomes this limitation. These temporal NMF models are then compared in a MAP framework on a prediction task, in the context of the Poisson likelihood.