Mathematical Pictures at a Data Science Exhibition

A diverse selection of data science topics explored through a mathematical lens.

Simon Foucart (Author)

9781009001854, Cambridge University Press

Paperback / softback, published 28 April 2022

350 pages
22.8 x 15.1 x 1.7 cm, 0.51 kg

'… an excellent discussion of representative algorithms as used in data science today - one of the best in-depth resources to appear in recent years for a scientist working on new analytic approaches or optimization … Highly recommended.' J. Brzezinski, Choice

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.

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.

Subject Areas: Neural networks & fuzzy systems [UYQN], Machine learning [UYQM], Mathematical theory of computation [UYA], Data capture & analysis [UNC], Optimization [PBU], Numerical analysis [PBKS]