Nonnegative Matrix Factorization for Signal and Data Analytics: Identifiability, Algorithms, and Applications

Nonnegative matrix factorization (NMF) aims at factoring a data matrix into low-rank latent factor matrices with nonnegativity constraints on (one or both of) the factors. Specifically, given a data matrix X ∈ RM×N and a target rank R, NMF seeks a factorization model X ≈WH>, W ∈ RM×R, H ∈ RN×R, to ‘explain’ the data matrix X, where W ≥ 0 and/or H ≥ 0 and R ≤ min{M,N}. At first glance, NMF is nothing but an alternative factorization model to existing ones such as the singular value decomposition (SVD) or independent component analysis (ICA) that have different constraints (resp. orthogonality or statistical independence) on the latent factors. However, the extraordinary effectiveness of NMF in analyzing real-life nonnegative data has sparked a substantial amount of research in many fields. The linear algebra community has shown interest in nonnegative matrices and nonnegative matrix factorization (known as nonnegative rank factorization) since more than thirty years ago [1]. In the 1990s, researchers in analytical chemistry and remote sensing (earth science) already noticed the effectiveness of NMF—which was first referred to as ‘positive matrix factorization’ [2, 3]. In 1999, Lee and Seung’s seminal paper published in Nature [4] sparked a tremendous amount of research on the computer science/signal processing side, since many reallife data like images, text, and audio spectra can be represented as nonnegative matrices—and