Maximum-Volume Nonnegative Matrix Factorization

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.

💡 Research Summary

This paper introduces a novel variant of nonnegative matrix factorization (NMF) that maximizes the volume of the coefficient matrix H, termed Maximum‑Volume NMF (MaxVol NMF). Traditional Minimum‑Volume NMF (MinVol NMF) seeks to shrink the volume of the basis matrix W, which can lead to two practical drawbacks: (i) low‑reflectance endmembers may be artificially suppressed, producing zero entries that have no physical meaning, and (ii) sparsity of H is only implicit, so noisy data often yield insufficiently sparse abundance maps.

The authors first establish a duality relationship: in the exact noiseless case, maximizing det(HHᵀ) is mathematically equivalent to minimizing det(WWᵀ). Under the same “sufficiently scattered” (SSC) condition on H, MaxVol NMF enjoys the same identifiability guarantees as MinVol NMF; any feasible solution is unique up to permutation.

In realistic noisy settings the authors propose the regularized objective
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