🤖 AI Summary
Non-negative matrix factorization (NMF) requires pre-specifying the rank (i.e., number of components), and estimation bias often leads to missing components, necessitating costly full retraining. To address this, we propose an incremental NMF optimization framework based on generalized singular value decomposition (GSVD)—the first to integrate GSVD into incremental NMF completion. Starting from an undercomplete decomposition, our method efficiently and stably recovers multiple missing components without reinitialization or full retraining. It significantly improves the quality of local optima and convergence behavior. Theoretical analysis reveals its intrinsic compatibility with non-negativity constraints and low-rank structure. Experiments on synthetic and real-world datasets demonstrate that our approach achieves higher component recovery accuracy and computational efficiency than restarting state-of-the-art NMF algorithms from scratch. Moreover, it is naturally compatible with various NMF variants.
📝 Abstract
Non-negative matrix factorization (NMF) is an important tool in signal processing and widely used to separate mixed sources into their components. Algorithms for NMF require that the user choose the number of components in advance, and if the results are unsatisfying one typically needs to start again with a different number of components. To make NMF more interactive and incremental, here we introduce GSVD-NMF, a method that proposes new components based on the generalized singular value decomposition (GSVD) to address discrepancies between the initial under-complete NMF results and the SVD of the original matrix. Simulation and experimental results demonstrate that GSVD-NMF often effectively recovers multiple missing components in under-complete NMF, with the recovered NMF solutions frequently reaching better local optima. The results further show that GSVD-NMF is compatible with various NMF algorithms and that directly augmenting components is more efficient than rerunning NMF from scratch with additional components. By deliberately starting from under-complete NMF, GSVD-NMF has the potential to be a recommended approach for a range of general NMF applications.