Least-squares methods for nonnegative matrix factorization over rational functions

TL;DR

This paper introduces rational-function constrained NMF (R-NMF), demonstrating its uniqueness and presenting three algorithms, with R-NMF outperforming traditional NMF in signal recovery and classification tasks.

Contribution

The paper establishes the uniqueness of R-NMF under mild conditions and compares three algorithms, highlighting trade-offs between speed and accuracy.

Findings

R-NMF has an essentially unique factorization unlike standard NMF.

R-HANLS is fast and accurate for large problems.

R-NMF outperforms NMF in signal recovery and classification.

Abstract

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.