A Second-Order Majorant Algorithm for Nonnegative Matrix Factorization

• URL: http://arxiv.org/abs/2303.17992v3
• Date: Wed, 18 Jun 2025 07:19:17 GMT
• Title: A Second-Order Majorant Algorithm for Nonnegative Matrix Factorization
• Authors: Mai-Quyen Pham, Jérémy Cohen, Thierry Chonavel,
• Abstract summary: We introduce a general second-order optimization framework for NMF under both quadratic and $beta$-divergence loss functions.<n>Second-Order Majorant (SOM) constructs a local quadratic majorization of the loss function by majorizing its Hessian matrix.<n>We show that mSOM consistently outperforms state-of-the-art algorithms across multiple loss functions.
• Score: 2.646309221150203
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Nonnegative Matrix Factorization (NMF) is a fundamental tool in unsupervised learning, widely used for tasks such as dimensionality reduction, feature extraction, representation learning, and topic modeling. Many algorithms have been developed for NMF, including the well-known Multiplicative Updates (MU) algorithm, which belongs to a broader class of majorization-minimization techniques. In this work, we introduce a general second-order optimization framework for NMF under both quadratic and $\beta$-divergence loss functions. This approach, called Second-Order Majorant (SOM), constructs a local quadratic majorization of the loss function by majorizing its Hessian matrix. It includes MU as a special case, while enabling faster variants. In particular, we propose mSOM, a new algorithm within this class that leverages a tighter local approximation to accelerate convergence. We provide a convergence analysis, showing linear convergence for individual factor updates and global convergence to a stationary point for the alternating version, AmSOM algorithm. Numerical experiments on both synthetic and real data sets demonstrate that mSOM consistently outperforms state-of-the-art algorithms across multiple loss functions.

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