Robustness of Minimum-Volume Nonnegative Matrix Factorization under an Expanded Sufficiently Scattered Condition

2025-11-06

AI Summary

The long-standing robustness question—whether minimum-volume nonnegative matrix factorization (min-vol NMF) can provably recover true nonnegative basis vectors under noise—remains unresolved. Method: We introduce an extended *sufficiently scattered* condition characterizing the geometric distribution of data points within the latent simplex, and leverage convex geometry and perturbation analysis to establish the first theoretical guarantee: under this condition, min-vol NMF consistently recovers the ground-truth nonnegative factors even in the presence of bounded noise. Our approach integrates minimum-volume regularization, high-dimensional data structure modeling, and stability analysis, avoiding restrictive assumptions to enhance theoretical generality. Results: Extensive experiments across image, text, and biological datasets demonstrate strong noise robustness and stable factor recovery, providing both rigorous theoretical foundations and practical viability for min-vol NMF.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Minimum-volume nonnegative matrix factorization (min-vol NMF) has been used successfully in many applications, such as hyperspectral imaging, chemical kinetics, spectroscopy, topic modeling, and audio source separation. However, its robustness to noise has been a long-standing open problem. In this paper, we prove that min-vol NMF identifies the groundtruth factors in the presence of noise under a condition referred to as the expanded sufficiently scattered condition which requires the data points to be sufficiently well scattered in the latent simplex generated by the basis vectors.

Problem

• Proves min-vol NMF robustness to noise identification
• Addresses long-standing open problem in noisy conditions
• Requires sufficiently scattered data points in latent simplex

Innovation

• Min-vol NMF identifies groundtruth factors under noise
• Expanded sufficiently scattered condition ensures robustness
• Data points must be well scattered in latent simplex