Low-Rank Factorization Models for Learning Biological Structure in Multi-Omics Data

High-throughput technologies have revolutionized the life sciences, leading to an unprecedented production of heterogeneous and high-dimensional datasets, collectively referred to as omics. These encompass genomics, transcriptomics, proteomics, metabolomics, epigenomics, and emerging domains such as microbiomics and radiomics. Each omics layer captures a distinct facet of biological complexity, yet no single modality can fully describe the intricate, multi-scale dynamics underlying health and disease. This realization has driven the development of integrative multi-omics models aimed at uncovering latent structures, merging heterogeneous data sources, and extracting actionable insights for translational medicine. Although these advances have opened remarkable opportunities, they have also introduced major computational challenges: omics data are typically high-dimensional, noisy, and sparse, with molecular features far exceeding available samples. Consequently, the search for methods capable of extracting reliable and interpretable knowledge from complex omics data has become increasingly pressing. Deep learning has marked a turning point in artificial intelligence and gained attention in biomedicine; however, these data-intensive and opaque architectures often lack interpretability and require large, labeled cohorts rarely accessible in biomedical research. In contexts such as oncology and precision medicine, where transparency is essential, such black-box models hinder clinical adoption, underscoring the need for approaches that pair analytical rigor with biological interpretability. Low-rank factorization models provide a principled and biologically grounded framework to disentangle high-dimensional omics signals into compact and interpretable latent structures aligned with domain knowledge. Methods such as nonnegative matrix factorization and tensor extensions decompose data into additive, part-based components often corresponding to coherent biological modules, with nonnegativity reflecting the nature of omics measurements and promoting interpretability. This dissertation advances the state of the art in low-rank factorizations for omics data analysis by introducing the semi-rigid orthogonal sparse Kullback–Leibler nonnegative matrix tri-factorization model and its multi-view extension. These approaches enhance stability, interpretability, and robustness in high-dimensional and noise-prone biomedical data, supporting key analytical objectives such as differential gene expression analysis and multi-omics integration. Novel initialization strategies and optimization procedures ensure reliable convergence and computational scalability, addressing longstanding limitations hindering the practical deployment of factorization models. Developed with a strong translational focus, the proposed frameworks are conceived as computational tools tailored to clinically relevant questions, complemented by additional integrative methods serving distinct analytical needs in precision oncology. Collectively, they enable patient stratification, biomarker discovery, and systematic characterization of tumor heterogeneity, thereby advancing the understanding of complex diseases and supporting precision medicine objectives. In conclusion, this work delivers methodological and translational contributions demonstrating that low-rank factorization constitutes a robust, interpretable, and clinically meaningful alternative to black-box models, paving the way for reliable computational tools in biomedical research and personalized oncology.