Recovering missing features in nonnegative matrix factorization via generalized singular value decomposition

Nonnegative matrix factorization (NMF) is widely used to separate mixed sources into components. Algorithms for NMF require choosing the rank in advance, and if the results are unsatisfying, one typically executes NMF again with a different rank. To make NMF more interactive, here we introduce GSVD-NMF, a method that proposes new components based on the generalized singular value decomposition (GSVD) to address discrepancies between 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. Furthermore, the under-complete NMF can be computed with a relaxed convergence tolerance, greatly reducing runtime while still enabling accurate feature recovery.