Mathematical and Statistical Foundations of Big Data Science: A Review of Methods and Challenges

Big Data Science has emerged as a transformative field driven by the rapid growth of large, complex, and high-dimensional datasets. This review examines the key mathematical and statistical principles that support the analysis, interpretation, and use of such data. In particular, it highlights the roles of linear algebra in data representation, probability theory in modeling uncertainty, optimization in large-scale computation, and statistical inference in drawing reliable conclusions. The review synthesizes existing studies into an integrated theoretical framework linking mathematical structure, statistical inference, and computational scalability. The literature was selected through a narrative review of publications indexed in Scopus, Web of Science, and Google Scholar, with a focus on studies published between 2005 and 2024. Relevant works were identified using keywords related to big data, mathematical foundations, statistical inference, and high-dimensional analysis. The review also discusses major challenges, including scalability, high dimensionality, data heterogeneity, noise, and limitations of traditional inferential methods. Finally, emerging approaches such as statistical learning, graph-based models, and the integration of mathematics with machine learning are highlighted as promising directions for future research.