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This paper introduces Maximum-Volume Nonnegative Matrix Factorization (MaxVol NMF), a dual approach to the standard technique that maximizes the volume of one factor matrix rather than minimizing it. The authors demonstrate that MaxVol NMF yields sparser, more interpretable decompositions and avoids rank-deficient solutions common in minimum-volume variants, offering improved performance for tasks like hyperspectral unmixing. Two new algorithms and a normalized variant are proposed to solve these optimization problems effectively.
Maximum-Volume Nonnegative Matrix Factorization
arXiv:2602.04795v3 Announce Type: replace Abstract: 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. The code is available from https://gitlab.com/vuthanho/maxvolmf.jl.
Abstract: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. The code is available from this https URL.
arXiv admin note: substantial text overlap with arXiv:2412.06380 (this paper is an updated version of Chapter 7 of the thesis of the first author, available from arXiv:2412.06380). The code is available from this https URL
Machine Learning (cs.LG); Signal Processing (eess.SP); Numerical Analysis (math.NA); Machine Learning (stat.ML)
arXiv:2602.04795 [cs.LG]
(or arXiv:2602.04795v3 [cs.LG] for this version)
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