MATHEMATICS FOR AI AND MACHINE LEARNING

Mathematics for AI and Machine Learning is a comprehensive, graduate-level textbook that provides the rigorous mathematical foundations essential for understanding modern artificial intelligence and machine learning systems.

The book spans 21 chapters organized into four parts. Part I (Chapters 1–10) covers linear algebra fundamentals: vector spaces, inner products, matrix operations, subspaces, orthogonality, QR decomposition, LU factorization, eigendecomposition, symmetric matrices, and the Singular Value Decomposition (SVD)—establishing the mathematical foundation for representation in AI.

Part II (Chapters 11–12) addresses differentiation and optimization: matrix calculus with gradients and Hessians, and optimization methods including gradient descent and its variants—formalizing learning as structured search in parameter space.

Part III (Chapters 13–16) introduces probability and information theory: probability and random variables, entropy and KL divergence, the Evidence Lower Bound (ELBO), variational inference and latent variable models, and Bellman equations for reinforcement learning—shifting the perspective from fitting functions to modeling distributions.

Part IV (Chapters 17–21) ventures into score functions, dynamics, and diffusion: score functions and energy-based models, Langevin dynamics and sampling methods, stochastic differential equations with Itô calculus, ODE/SDE continuous limits of algorithms, and Fokker-Planck equations governing distribution dynamics—framing generative modeling as the study of distributional dynamics.

What distinguishes this textbook is its seamless integration of mathematical rigor with practical AI/ML applications. Each concept is motivated by real-world problems in machine learning, deep learning, large language models, graph neural networks, reinforcement learning, and modern generative frameworks. The full-color figures illuminate complex ideas, while extensive exercises reinforce understanding.

Designed for graduate students, researchers, and experienced practitioners, this book serves as both a learning resource and a comprehensive reference. Whether you're building foundation models, researching novel architectures, or seeking deeper understanding of the mathematics powering AI systems, this textbook provides the essential theoretical toolkit.

Fuheng Wu is a Principal ML Tech Lead at Oracle Generative AI, specializing in distributed training, inference, and GPU systems for enterprise AI workloads. He delivered core components of Oracle's large-scale computer-vision, document-AI models, and Code Assist(Copilot), and co-authored a Microsoft-Oracle blog on deep learning. An alumnus of the Singapore-MIT Alliance, where he studied Linear Algebra under Gilbert Strang, he has worked at NetEase and Uber and contributed to open-source AI projects including SGLang, genai-bench, pyLLaMA, chatLLaMA, HiQ and AthenaDriver. He is the author of Mathematics for AI and Machine Learning. He teaches Generative AI foundations at the McCombs School of Business, University of Texas at Austin, and mentors students preparing for AI Olympiads. Writing as Xuan Xin, he is also the author of the poetic memoir Above the Clouds and the music album Let's Train The Model (2024), which renders deep learning and AI concepts as music. Beyond technology, he is a Zen calligrapher and volunteer math tutor.