  CMF is a technique for simultaneously learning low-rank representations based
on a collection of matrices with shared entities. A typical example is the
joint modeling of user-item, item-property, and user-feature matrices in a
recommender system. The key idea in CMF is that the embeddings are shared
across the matrices, which enables transferring information between them. The
existing solutions, however, break down when the individual matrices have
low-rank structure not shared with others. In this work we present a novel CMF
solution that allows each of the matrices to have a separate low-rank structure
that is independent of the other matrices, as well as structures that are
shared only by a subset of them. We compare MAP and variational Bayesian
solutions based on alternating optimization algorithms and show that the model
automatically infers the nature of each factor using group-wise sparsity. Our
approach supports in a principled way continuous, binary and count observations
and is efficient for sparse matrices involving missing data. We illustrate the
solution on a number of examples, focusing in particular on an interesting
use-case of augmented multi-view learning.
