Chem-NMF: Multi-layer $α$-divergence Non-Negative Matrix Factorization for Cardiorespiratory Disease Clustering, with Improved Convergence Inspired by Chemical Catalysts and Rigorous Asymptotic Analysis

2025-10-08

AI Summary

To address the lack of theoretical convergence guarantees for multilayer α-divergence nonnegative matrix factorization (NMF) in cardiopulmonary disease clustering, this paper proposes Chem-NMF—a novel multilayer NMF framework inspired by chemical catalysis mechanisms. Its core innovation lies in the first incorporation of the Boltzmann probability model into NMF convergence analysis, enabling a theoretically grounded, stable convergence mechanism that jointly leverages energy barriers and boundary factors to ensure asymptotic convergence of multilayer α-divergence NMF. The method integrates α-divergence optimization, nonnegativity constraints, deep hierarchical architecture, and rigorous asymptotic convergence analysis, making it particularly suitable for biomedical signals and images. Experiments demonstrate significant improvements: +5.6% ± 2.7% in clustering accuracy on cardiopulmonary disease signal data and +11.1% ± 7.2% on face image data, substantially outperforming existing NMF variants.

Technology Category

• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Data Compression
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Non-Negative Matrix Factorization (NMF) is an unsupervised learning method offering low-rank representations across various domains such as audio processing, biomedical signal analysis, and image recognition. The incorporation of $α$-divergence in NMF formulations enhances flexibility in optimization, yet extending these methods to multi-layer architectures presents challenges in ensuring convergence. To address this, we introduce a novel approach inspired by the Boltzmann probability of the energy barriers in chemical reactions to theoretically perform convergence analysis. We introduce a novel method, called Chem-NMF, with a bounding factor which stabilizes convergence. To our knowledge, this is the first study to apply a physical chemistry perspective to rigorously analyze the convergence behaviour of the NMF algorithm. We start from mathematically proven asymptotic convergence results and then show how they apply to real data. Experimental results demonstrate that the proposed algorithm improves clustering accuracy by 5.6% $pm$ 2.7% on biomedical signals and 11.1% $pm$ 7.2% on face images (mean $pm$ std).

Problem

• Improving convergence in multi-layer NMF architectures
• Applying chemical catalyst theory to analyze algorithm convergence
• Enhancing clustering accuracy for biomedical and image data

Innovation

• Multi-layer α-divergence NMF for disease clustering
• Chemical catalyst-inspired convergence stabilization method
• Rigorous asymptotic analysis for convergence behavior