Maximum-Volume Nonnegative Matrix Factorization

2026-02-04

AI Summary

This work proposes MaxVol NMF, a novel nonnegative matrix factorization framework that maximizes the volume of the coefficient matrix \( H \), addressing the non-uniqueness and poor interpretability of traditional NMF and the rank-deficient solutions often produced by minimum-volume NMF under noise. By maximizing volume, the method ensures identifiability in noise-free settings and reveals that its solution corresponds to an exclusive clustering of data columns. A normalized variant is further introduced, bridging standard NMF and orthogonal NMF. Experimental results on hyperspectral unmixing demonstrate that MaxVol NMF significantly outperforms MinVol NMF, effectively avoiding rank deficiency and yielding sparser, structurally clearer, and more interpretable decompositions.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Other Foundations of Computer Vision
• Search and Optimization: Non-convex Optimization

Application Category

• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

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.

Problem

• Nonnegative Matrix Factorization
• Maximum-Volume
• Identifiability
• Sparsity
• Hyperspectral Unmixing

Innovation

• Maximum-Volume NMF
• Nonnegative Matrix Factorization
• Identifiability
• Sparse Decomposition
• Hyperspectral Unmixing