Mathematical pictures at a data science exhibition
- United Kingdom: Cambridge University Press, 2022.
- xx, 318p.; 22cms.
This text provides deep and comprehensive coverage of the mathematical background for data science, including machine learning, optimal recovery, compressed sensing, optimization, and neural networks. In the past few decades, heuristic methods adopted by big tech companies have complemented existing scientific disciplines to form the new field of Data Science. This text embarks the readers on an engaging itinerary through the theory supporting the field. Altogether, twenty-seven lecture-length chapters with exercises provide all the details necessary for a solid understanding of key topics in data science. While the book covers standard material on machine learning and optimization, it also includes distinctive presentations of topics such as reproducing kernel Hilbert spaces, spectral clustering, optimal recovery, compressed sensing, group testing, and applications of semidefinite programming. Students and data scientists with less mathematical background will appreciate the appendices that provide more background on some of the more abstract concepts.
Specially designed for mathematicians and graduate students in mathematics who want to learn more about data science Presents a broad view of mathematical data science by including a wide variety of subjects, from the very popular subject of machine learning to the lesser-known subject of optimal recovery Proves at least one theoretical result in each chapter, helping the reader develop a sound understanding of topics explained with detailed arguments Includes original content that has never been published before in book form, such as the presentation of compressive sensing through a nonstandard restricted isometry property Provides background for some of the more abstract concepts in the appendices Author's GitHub page includes computational illustrations made in MATLAB and Python to demonstrate how the theory is applied
Part I. Machine Learning:
1. Rudiments of Statistical Learning 2. Vapnik–Chervonenkis Dimension 3. Learnability for Binary Classification 4. Support Vector Machines 5. Reproducing Kernel Hilbert 6. Regression and Regularization 7. Clustering 8. Dimension Reduction Part II Optimal Recovery: 9. Foundational Results of Optimal Recovery 10. Approximability Models 11. Ideal Selection of Observation Schemes 12. Curse of Dimensionality 13. Quasi-Monte Carlo Integration Part III Compressive Sensing: 14. Sparse Recovery from Linear Observations 15. The Complexity of Sparse Recovery 16. Low-Rank Recovery from Linear Observations 17. Sparse Recovery from One-Bit Observations 18. Group Testing Part IV Optimization: 19. Basic Convex Optimization 20. Snippets of Linear Programming 21. Duality Theory and Practice 22. Semidefinite Programming in Action 23. Instances of Nonconvex Optimization Part V Neural Networks: 24. First Encounter with ReLU Networks 25. Expressiveness of Shallow Networks 26. Various Advantages of Depth 27. Tidbits on Neural Network Training Appendix A High-Dimensional Geometry Appendix B. Probability Theory Appendix C. Functional Analysis Appendix D. Matrix Analysis Appendix E. Approximation Theory
9781009001854
Computer science--Mathematics Compressed sensing, optimization, and neural networks Rudiments of Statistical Learning