Matrix Factorization: An Introduction to Traditional Methods and a Review of Recent Application Research

Abstract

Matrix decomposition, as a core mathematical tool for low-rank representation and latent feature extraction in high-dimensional data, has evolved in recent years from the realm of classical numeri-cal linear algebra to become a key component of intelligent computing engines. This paper systemat-ically reviews the mathematical mechanisms, stability conditions, and algorithmic complexity of three major classical methods: LU, Cholesky, and SVD. It identifies theoretical bottlenecks within the traditional framework concerning interpretability, computational efficiency, and complex pattern characterization. Subsequently, it focuses on three interdisciplinary scenarios—hyperspectral remote sensing, lncRNA–disease association prediction, and ground-penetrating radar clutter suppres-sion—to dissect recent advancements and failure mechanisms in non-negative matrix factorization, low-rank sparse modelling, and tensor regularization strategies. Furthermore, it proposes a synergis-tic optimization pathway integrating domain knowledge, regularization constraints, and stochastic algorithms. This demonstrates the feasibility of embedding physical equations, biological priors, and deep learning nonlinear representations into a unified loss function, enabling a paradigm shift from static data analysis to dynamic system intervention. Finally, we envision personalized computational models for millisecond-scale closed-loop neural modulation, providing an extensible methodological framework for deep applications of matrix decomposition in precision medicine and real-time engi-neering systems.