  Nonnegative Matrix Factorization (NMF) aims to factorize a matrix into two
optimized nonnegative matrices and has been widely used for unsupervised
learning tasks such as product recommendation based on a rating matrix.
However, although networks between nodes with the same nature exist, standard
NMF overlooks them, e.g., the social network between users. This problem leads
to comparatively low recommendation accuracy because these networks are also
reflections of the nature of the nodes, such as the preferences of users in a
social network. Also, social networks, as complex networks, have many different
structures. Each structure is a composition of links between nodes and reflects
the nature of nodes, so retaining the different network structures will lead to
differences in recommendation performance. To investigate the impact of these
network structures on the factorization, this paper proposes four multi-level
network factorization algorithms based on the standard NMF, which integrates
the vertical network (e.g., rating matrix) with the structures of horizontal
network (e.g., user social network). These algorithms are carefully designed
with corresponding convergence proofs to retain four desired network
structures. Experiments on synthetic data show that the proposed algorithms are
able to preserve the desired network structures as designed. Experiments on
real-world data show that considering the horizontal networks improves the
accuracy of document clustering and recommendation with standard NMF, and
various structures show their differences in performance on these two tasks.
These results can be directly used in document clustering and recommendation
systems.
