Maximum-Volume Nonnegative Matrix Factorization
This work addresses the interpretability and uniqueness challenges in NMF for data embedding, particularly in domains like hyperspectral unmixing, but it is incremental as it builds on the established minimum-volume NMF framework.
The paper tackles the problem of nonnegative matrix factorization (NMF) by proposing a maximum-volume approach (MaxVol NMF) that maximizes the volume of the factor H, which is more effective for sparse decomposition and avoids rank-deficient solutions compared to the existing minimum-volume method. The results show that MaxVol NMF extracts sparse decompositions and clusters columns of the data matrix, with a normalized variant outperforming both standard and orthogonal NMF in hyperspectral unmixing.
Nonnegative matrix factorization (NMF) is a popular data embedding technique. Given a nonnegative data matrix $X$, it aims at finding two lower dimensional matrices, $W$ and $H$, such that $X\approx WH$, where the factors $W$ and $H$ are constrained to be element-wise nonnegative. The factor $W$ serves as a basis for the columns of $X$. In order to obtain more interpretable and unique solutions, minimum-volume NMF (MinVol NMF) minimizes the volume of $W$. In this paper, we consider the dual approach, where the volume of $H$ is maximized instead; this is referred to as maximum-volume NMF (MaxVol NMF). MaxVol NMF is identifiable under the same conditions as MinVol NMF in the noiseless case, but it behaves rather differently in the presence of noise. In practice, MaxVol NMF is much more effective to extract a sparse decomposition and does not generate rank-deficient solutions. In fact, we prove that the solutions of MaxVol NMF with the largest volume correspond to clustering the columns of $X$ in disjoint clusters, while the solutions of MinVol NMF with smallest volume are rank deficient. We propose two algorithms to solve MaxVol NMF. We also present a normalized variant of MaxVol NMF that exhibits better performance than MinVol NMF and MaxVol NMF, and can be interpreted as a continuum between standard NMF and orthogonal NMF. We illustrate our results in the context of hyperspectral unmixing.