Non-Negative Matrix Factorization with Auxiliary Information on Overlapping Groups

Matrix factorization is useful to extract the essential low-rank structure from a given matrix and has been paid increasing attention. A typical example is non-negative matrix factorization (NMF), which is one type of unsupervised learning, having been successfully applied to a variety of data including documents, images and gene expression, where their values are usually non-negative. We propose a new model of NMF which is trained by using auxiliary information of overlapping groups. This setting is very reasonable in many applications, a typical example being gene function estimation where functional gene groups are heavily overlapped with each other. To estimate true groups from given overlapping groups efficiently, our model incorporates latent matrices with the regularization term using a mixed norm. This regularization term allows group-wise sparsity on the optimized low-rank structure. The latent matrices and other parameters are efficiently estimated by a block coordinate gradient descent method. We empirically evaluated the performance of our proposed model and algorithm from a variety of viewpoints, comparing with four methods including MMF for auxiliary graph information, by using both synthetic and real world document and gene expression data sets.