Least-Squares Methods for Nonnegative Matrix Factorization Over Rational Functions

Nonnegative matrix factorization (NMF) models are widely used to analyze 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. This leads to a fairly general model referred to as NMF using rational functions (R-NMF). We first show that, unlike NMF, R-NMF possesses an essentially unique factorization under mild assumptions, which is crucial in applications where the ground-truth factors need to be recovered, as in blind source separation problems. Then we present different approaches to solve R-NMF: the R-HANLS, R-ANLS and R-NLS methods. In our tests, no method significantly outperforms the others in all cases, and all three methods offer a different trade-off between solution accuracy and computational requirements. Indeed, while R-HANLS is fast and accurate for large problems, R-ANLS is more accurate, but also more resources demanding, both in time and memory and R-NLS is even more accurate but only for small problems. Then, crucially we show that R-NMF models outperforms NMF in various tasks including the recovery of semi-synthetic continuous signals, and a classification problem of real hyperspectral signals.