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.
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