  A mixed data frame (MDF) is a table collecting categorical, numerical and
count observations. The use of MDF is widespread in statistics and the
applications are numerous from abundance data in ecology to recommender
systems. In many cases, an MDF exhibits simultaneously main effects, such as
row, column or group effects and interactions, for which a low-rank model has
often been suggested. Although the literature on low-rank approximations is
very substantial, with few exceptions, existing methods do not allow to
incorporate main effects and interactions while providing statistical
guarantees. The present work fills this gap.
  We propose an estimation method which allows to recover simultaneously the
main effects and the interactions. We show that our method is near optimal
under conditions which are met in our targeted applications. We also propose an
optimization algorithm which provably converges to an optimal solution.
Numerical experiments reveal that our method, mimi, performs well when the main
effects are sparse and the interaction matrix has low-rank. We also show that
mimi compares favorably to existing methods, in particular when the main
effects are significantly large compared to the interactions, and when the
proportion of missing entries is large. The method is available as an R package
on the Comprehensive R Archive Network.
