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.
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.
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.
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.
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.