Integrative Analysis of Three-Dimensional Data Arrays with Non-negative Matrix Factorization

Collections of data matrices (data arrays) are becoming increasingly available due to advances in acquisition technologies. Various matrix factorization techniques have been applied to such data for integrating them and extraction of common patterns. In this paper, we investigate citation networks – networks of references (edges) between documents (nodes) – and find that existing methods rely on a direct application of such techniques and can be adversely affected by transient patterns, and thus fail to extract parsimonious ones. We introduce a model for non-negative matrix factorization that captures how structure evolves through the data. The model decomposes the data into two factors: a basis common to all data matrices, and a coefficient matrix that varies for each data matrix. A regularization is utilized within the framework of non-negative matrix factorization to encourage local smoothness of the coefficient matrix. This improves interpretability and highlights the structural patterns underlying the data, while mitigating noise effects. To demonstrate the generality of the proposed methodology, it is also illustrated on bilateral trade data that features space-time dynamics different from typical citation networks.