Group Sparsity in Nonnegative Matrix Factorization

A recent challenge in data analysis for science and engineering is that data are often represented in a structured way. In particular, many data mining tasks have to deal with group-structured prior information, where features or data items are organized into groups. In this paper, we develop group sparsity regularization methods for nonnegative matrix factorization (NMF). NMF is an effective data mining tool that has been widely adopted in text mining, bioinformatics, and clustering, but a principled approach to incorporating group information into NMF has been lacking in the literature. Motivated by an observation that features or data items within a group are expected to share the same sparsity pattern in their latent factor representation, we propose mixednorm regularization to promote group sparsity in the factor matrices of NMF. Group sparsity improves the interpretation of latent factors. Efficient convex optimization methods for dealing with the mixed-norm term are presented along with computational comparisons between them. Application examples of the proposed method in factor recovery, semi-supervised clustering, and multilingual text analysis are demonstrated.