Algebra, Topology & Optimization for Machine Learning by Jean Gallier

About this book :-
"Algebra, Topology, Differential Calculus, and Optimization Theory for Computer Science and Machine Learning" by Jean Gallier and Jocelyn Quaintance is a detailed guide bridging "mathematics", "machine learning", and "algorithm design". The book focuses on the essential mathematical tools behind modern data-driven technologies, making it an indispensable resource for students, researchers, and professionals in "artificial intelligence" and "data science". The first part of the book explores "linear algebra" in depth, covering vector spaces, matrices, eigenvalues, and linear transformations. Gallier emphasizes how these concepts are directly applied in machine learning tasks such as feature extraction, dimensionality reduction, and model representation. Clear examples and illustrations help readers understand the connection between theory and practical computation, ensuring a strong foundation in the mathematics behind algorithms. In the later sections, Gallier delves into "topology" and "optimization", explaining manifolds, metric spaces, and convex optimization problems. These topics are critical for understanding neural networks, manifold learning, and optimization-based learning algorithms. By combining rigorous theory with real-world applications, the book enables readers to analyze and design efficient machine learning models. Its structured approach ensures that even complex topics are accessible, providing the mathematical insight necessary to advance in "data science" and "AI" fields.

Book Detail :-
Title: Algebra, Topology & Optimization for Machine Learning by Jean Gallier
Publisher: World Scientific Publishing Co.
Year: 2025
Pages: 2204
Type: PDF
Language: English
ISBN-10 #: 110845514X
ISBN-13 #: 978-1108455145
License: University Resource
Amazon: Amazon

About Author :-
The author Jean Gallier and Jocelyn Quaintance is a renowned "mathematician" and "educator" specializing in "algebra", "topology", and "optimization". He bridges abstract mathematical concepts with practical applications, making complex topics accessible to students and professionals in "machine learning". With extensive research and teaching experience, Gallier focuses on how "algebraic structures", topological insights, and optimization techniques underpin modern computational methods. His work emphasizes clarity, rigor, and real-world applications, helping learners understand both theory and practice in "data science" and AI. Gallier’s contributions continue to influence the mathematical foundations of contemporary machine learning and algorithm design.

Book Contents :-
INTRODUCTION 1. Introduction 2. Groups, Rings, and Fields PART-I LINEAR ALGEBRA 3. Vector Spaces, Bases, Linear Maps 4. Matrices and Linear Maps 5. Haar Bases, Haar Wavelets, Hadamard Matrices 6. Direct Sums 7. Determinants 8. Gaussian Elimination, LU, Cholesky, Echelon Form 9. Vector Norms and Matrix Norms 10. Iterative Methods for Solving Linear Systems 11. The Dual Space and Duality 12. Euclidean Spaces 13. QR-Decomposition for Arbitrary Matrices 14. Hermitian Spaces 15. Eigenvectors and Eigenvalues 16. Unit Quaternions and Rotations in SO(3) 17. Spectral Theorems 18. Computing Eigenvalues and Eigenvectors 19. Introduction to the Finite Element Method 20. Graphs and Graph Laplacians 21. Spectral Graph Drawing 22. Singular Value Decomposition and Polar Form 23. Applications of SVD and Pseudo-Inverses PART-II AFFINE AND PROJECTIVE GEOMETRY 24. Basics of Affine Geometry 25. Embedding an Affine Space in a Vector Space 26. Basics of Projective Geometry PART-III THE GEOMETRY OF BILINEAR FORMS 27. The Cartan–Dieudonné Theorem 28. Isometries of Hermitian Spaces 29. The Geometry of Bilinear Forms. Witt’s Theorem PART-IV ALGEBRA. PIDS, UFDS, NOETHERIAN RINGS, TENSORS 30. Polynomials, Ideals, and PIDs 31. Annihilating Polynomials and Primary Decomposition 32. UFDs, Noetherian Rings, Hilbert’s Basis Theorem 33. Tensor Algebras 34. Exterior Tensor Powers and Exterior Algebras 35. Introduction to Modules. Modules over a PID 36. Normal Forms and the Rational Canonical Form PART-V TOPOLOGY AND DIFFERENTIAL CALCULUS 37. Topology 38. A Detour on Fractals 39. Differential Calculus PART-VI PRELIMINARIES FOR OPTIMIZATION THEORY 40. Extrema of Real-Valued Functions 41. Newton’s Method and Its Generalizations 42. Quadratic Optimization Problems 43. Schur Complements and Applications PART-VII LINEAR OPTIMIZATION 44. Convex Sets, Cones, H-Polyhedra 45. Linear Programs 46. The Simplex Algorithm 47. Linear Programming and Duality PART-VIII NONLINEAR OPTIMIZATION 48. Basics of Hilbert Spaces 49. General Results of Optimization Theory

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