Mathematics of Data Science

2026-07-11

AI Summary

This work addresses the theoretical challenges posed by high-dimensional data by establishing a unified mathematical foundation for data science. Integrating tools from high-dimensional probability, matrix analysis, spectral graph theory, and optimization, it systematically elucidates the mathematical principles underlying core algorithms such as dimensionality reduction, regression, clustering, classification, deep learning, and sparse recovery. Key techniques include singular value decomposition, principal component analysis, random projections, diffusion maps, regularization methods, graph Laplacian limits, and compressed sensing. A rigorous analytical framework is developed using matrix concentration inequalities and related probabilistic tools. The resulting theory constitutes a coherent and self-contained system that provides solid theoretical support for the design and performance analysis of data science algorithms.

Abstract

This book is about the mathematical foundations of data science. 1. Introduction 2. Curses, Blessings, and Surprises in High Dimensions 3. Singular Value Decomposition and Principal Component Analysis 4. Linear Regression and Regularization 5. Graphs, Networks, and Clustering 6. Nonlinear Dimension Reduction and Diffusion Maps 7. Linear Dimension Reduction via Random Projections 8. Optimization for Data Science 9. Classification 10. A Mathematical Introduction to Deep Learning 11. Large Sample Limit of Graph Laplacians 12. Community 13. Concentration of Measure and Gaussian Analysis 14. Matrix Concentration Inequalities 15. Compressive Sensing and Sparsity 16. Low-Rank Matrix Recovery

Problem

• Data Science
• Mathematical Foundations
• High-Dimensional Phenomena
• Dimensionality Reduction
• Matrix Recovery

Innovation

• High-dimensional analysis
• Dimensionality reduction
• Matrix concentration inequalities
• Compressive sensing
• Graph Laplacians