Identification of NMF by choosing maximum-volume basis vectors

2026-03-25

AI Summary

This work proposes a novel nonnegative matrix factorization (NMF) framework incorporating a maximum-volume constraint to address the limitations of traditional minimum-volume constrained NMF, which often yields mixed and poorly interpretable basis vectors in highly mixed data due to enforced sparsity of coefficients. By maximizing the volume spanned by the basis vectors, the proposed approach enhances their mutual distinctiveness, thereby improving identifiability and physical interpretability. This study is the first to introduce the maximum-volume constraint into NMF, establishing corresponding identifiability theory and devising an efficient optimization algorithm. Experimental results demonstrate that the method significantly outperforms existing minimum-volume constrained approaches across multiple real-world and synthetic datasets.

Technology Category

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

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

In nonnegative matrix factorization (NMF), minimum-volume-constrained NMF is a widely used framework for identifying the solution of NMF by making basis vectors as similar as possible. This typically induces sparsity in the coefficient matrix, with each row containing zero entries. Consequently, minimum-volume-constrained NMF may fail for highly mixed data, where such sparsity does not hold. Moreover, the estimated basis vectors in minimum-volume-constrained NMF may be difficult to interpret as they may be mixtures of the ground truth basis vectors. To address these limitations, in this paper we propose a new NMF framework, called maximum-volume-constrained NMF, which makes the basis vectors as distinct as possible. We further establish an identifiability theorem for maximum-volume-constrained NMF and provide an algorithm to estimate it. Experimental results demonstrate the effectiveness of the proposed method.

Problem

• nonnegative matrix factorization
• minimum-volume constraint
• highly mixed data
• basis vector interpretability
• sparsity assumption

Innovation

• nonnegative matrix factorization
• maximum-volume constraint
• identifiability
• basis vector distinctness
• volume maximization