SGRiT: Non-Negative Matrix Factorization via Subspace Graph Regularization and Riemannian-Based Trust Region Algorithm

Non-negative Matrix Factorization (NMF) has gained popularity due to its effectiveness in clustering and feature selection tasks. It is particularly valuable for managing high-dimensional data by reducing dimensionality and providing meaningful semantic representations. However, traditional NMF methods may encounter challenges when dealing with noisy data, outliers, or when the underlying manifold structure of the data is overlooked. This paper introduces an innovative approach called SGRiT, which employs Stiefel manifold optimization to enhance the extraction of latent features. These learned features have been shown to be highly informative for clustering tasks. The method leverages a spectral decomposition criterion to obtain a low-dimensional embedding that captures the intrinsic geometric structure of the data. Additionally, this paper presents a solution for addressing the Stiefel manifold problem and utilizes a Riemannian-based trust region algorithm to optimize the loss function. The outcome of this optimization process is a new representation of the data in a transformed space, which can subsequently serve as input for the NMF algorithm. Furthermore, this paper incorporates a novel subspace graph regularization term that considers high-order geometric information and introduces a sparsity term for the factor matrices. These enhancements significantly improve the discrimination capabilities of the learning process. This paper conducts an impartial analysis of several essential NMF algorithms. To demonstrate that the proposed approach consistently outperforms other benchmark algorithms, four clustering evaluation indices are employed.