MATHEMATICS FOR AI AND MACHINE LEARNING: A Comprehensive Mathematical Reference for Artificial Intelligence 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 and document AI models, as well as Oracle Code Assist, and co-authored a Microsoft-Oracle technical blog on deep learning. An alumnus of the Singapore-MIT Alliance, where he studied linear algebra under Gilbert Strang, Wu also earned an M.Sc. in Financial Engineering from the National University of Singapore, where he studied stochastic differential equations under Steven Kou. Earlier in his career, he worked in NetEase's President's Office, reporting directly to NetEase founder William Ding, and later worked at Uber. He has 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 has taught generative AI in a program associated with the McCombs School of Business at the University of Texas at Austin and mentored 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.

Yu Guo is Vice President of Data, Analytics, and AI at BP, where he leads work at the intersection of artificial intelligence, data, and enterprise technology. He holds a PhD in Biostatistics from Harvard University and has held leadership roles in machine learning, data science, and engineering at Uber, Airbnb, Microsoft, Aurora, and TomTom.