  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.
