Nonnegative Matrix Factorization with Band Constraint

Abstract

In this paper, we study a band constrained nonnegative matrix factorization (band NMF) problem: for a given nonnegative matrix $Y$, decompose it as $Y ≈ AX$ with $A$ a nonnegative matrix and $X$ a nonnegative block band matrix. This factorization model extends a single low rank subspace model to a mixture of several overlapping low rank subspaces, which not only can provide sparse representation, but also can capture significant grouping structure from a dataset. Based on overlapping subspace clustering and the capture of the level of overlap between neighbouring subspaces, two simple and practical algorithms are presented to solve the band NMF problem. Numerical experiments on both synthetic data and real images data show that band NMF enhances the performance of NMF in data representation and processing.