Penalizing Low-Rank Matrix Factorization: From theoretical connections to practical applications

Low-rank (LR) factorization techniques aim to represent data in a low-dimensional space by identifying fundamental sources. Standard LR approaches often require additional constraints to account for real-world complexity, resulting in penalized low-rank matrix factorizations. These techniques incorporate penalties or regularization terms to improve robustness and adaptability to practical constraints, bridging theoretical research with real-world applications. This paper explores a nonnegative constrained low-rank decomposition technique, namely, Nonnegative Matrix Factorization (NMF), and its constrained variants as powerful tools for analyzing nonnegative data. We cover theoretical foundations and practical implementations, review algorithms for standard NMF, and address challenges in setting hyperparameters for penalized variants. We emphasize applications in omics data analysis with a model that incorporates biological constraints to extract meaningful insights, and highlight applications in environmental data analysis