Semi-supervised symmetric non-negative matrix factorization with graph quality improvement and constraints

Symmetric non-negative matrix factorization (SNMF) decomposes a similarity matrix into the product of an indicator matrix and its transpose, allowing clustering results to be directly extracted from the indicator matrix without additional clustering methods. Furthermore, SNMF has been shown to be effective in clustering nonlinearly separable data. SNMF-based clustering methods significantly depend on the quality of the pairwise similarity matrix, yet their effectiveness is often hindered by the reliance on predefined matrices in most semi-supervised SNMF approaches. Thus, we propose a novel algorithm, named semi-supervised symmetric non-negative matrix factorization with graph quality improvement and constraints (S3NMFGC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {S}<^>{3}\text {NMFGC}$$\end{document}), addressing this limitation by employing an integrated clustering strategy that dynamically generates and adaptively updates the similarity matrices. This is accomplished by integrating a weighted graph construction based on multiple clustering results, a label propagation algorithm, and pairwise constraint terms into a unified optimization framework that enhances the semi-supervised SNMF model. Subsequently, we adopt an alternating iterative update method to solve the optimization problem and prove its convergence. Rigorous experiments highlight the superiority of our model, which outperforms seven state-of-the-art NMF methods across six datasets.