Multiplicative update rules for nonnegative matrix factorization with co-occurrence constraints

Nonnegative matrix factorization (NMF) is a widely-used tool for obtaining low-rank approximations of nonnegative data such as digital images, audio signals, textual data, financial data, and more. One disadvantage of the basic NMF formulation is its inability to control the amount of dependence among the learned dictionary atoms. Enforcing dependence within predetermined groups of atoms allows objects to be represented using multiple atoms instead of only one atom. In this paper, we introduce three simple and convenient multiplicative update rules for NMF that enforce dependence among atoms. Using examples in music transcription, we demonstrate the ability of these updates to represent each musical note with multiple atoms and cluster the atoms for source separation purposes.