The Mathematical Foundations for Data Representation and Analysis in Data Science: A Comprehensive Review

The rapid advancement of data-driven technologies has underscored the importance of mathematical foundations in data science, particularly in how data is represented and analyzed. This review delves into the key mathematical frameworks that support modern data science, focusing on areas like linear algebra, probability, optimization, and topology. These mathematical tools form the backbone for efficiently representing data, discovering patterns, and constructing predictive models. The review explores techniques such as dimensionality reduction, sparse representations, and manifold learning, highlighting both their theoretical bases and practical uses. It also discusses challenges encountered in large-scale and complex data, such as scalability, data quality, and interpretability. By summarizing recent developments and identifying unresolved issues, the review seeks to offer a thorough understanding of the mathematical principles that drive progress in data science methods and applications.