Least-squares methods for nonnegative matrix factorization over rational functions
This work addresses the need for unique factor recovery in blind source separation and similar applications, but it is incremental as it builds on existing NMF methods with rational constraints.
The paper tackled the problem of nonnegative matrix factorization for continuous signals by constraining factors to rational functions (R-NMF), showing it has essentially unique factorization unlike standard NMF, which aids in applications like blind source separation. They presented three methods (R-HANLS, R-ANLS, R-NLS) with trade-offs in speed and accuracy, and demonstrated that R-NMF outperforms NMF in tasks such as recovering semi-synthetic signals and classifying real hyperspectral signals.
Nonnegative Matrix Factorization (NMF) models are widely used to recover linearly mixed nonnegative data. When the data is made of samplings of continuous signals, the factors in NMF can be constrained to be samples of nonnegative rational functions, which allow fairly general models; this is referred to as NMF using rational functions (R-NMF). We first show that, under mild assumptions, R-NMF has an essentially unique factorization unlike NMF, which is crucial in applications where ground-truth factors need to be recovered such as blind source separation problems. Then we present different approaches to solve R-NMF: the R-HANLS, R-ANLS and R-NLS methods. From our tests, no method significantly outperforms the others, and a trade-off should be done between time and accuracy. Indeed, R-HANLS is fast and accurate for large problems, while R-ANLS is more accurate, but also more resources demanding, both in time and memory. R-NLS is very accurate but only for small problems. Moreover, we show that R-NMF outperforms NMF in various tasks including the recovery of semi-synthetic continuous signals, and a classification problem of real hyperspectral signals.