  Count data are collected in many scientific and engineering tasks including
image processing, single-cell RNA sequencing and ecological studies. Such data
sets often contain missing values, for example because some ecological sites
cannot be reached in a certain year. In addition, in many instances, side
information is also available, for example covariates about ecological sites or
species. Low-rank methods are popular to denoise and impute count data, and
benefit from a substantial theoretical background. Extensions accounting for
covariates have been proposed, but to the best of our knowledge their
theoretical and empirical properties have not been thoroughly studied, and few
softwares are available for practitioners. We propose a complete methodology
called LORI (Low-Rank Interaction), including a Poisson model, an algorithm,
and automatic selection of the regularization parameter, to analyze count
tables with covariates. We also derive an upper bound on the estimation error.
We provide a simulation study with synthetic data, revealing empirically that
LORI improves on state of the art methods in terms of estimation and imputation
of the missing values. We illustrate how the method can be interpreted through
visual displays with the analysis of a well-know plant abundance data set, and
show that the LORI outputs are consistent with known results. Finally we
demonstrate the relevance of the methodology by analyzing a water-birds
abundance table from the French national agency for wildlife and hunting
management (ONCFS). The method is available in the R package lori on the
Comprehensive Archive Network (CRAN).
