This book provides a systematic introduction to the mathematical structures underlying modern machine learning and artificial intelligence. It develops the functional, metric, algebraic, and optimization tools needed to formulate learning problems, analyze algorithms, and interpret model behavior beyond purely computational implementations.

The exposition begins with metric spaces, normed spaces, Hilbert spaces, reproducing kernel Hilbert spaces, operators, and fixed-point methods, emphasizing their role in data representation and learning. It then reviews the essential elements of linear algebra, including eigenanalysis, singular value decomposition, pseudoinverses, and matrix conditioning. A substantial part of the volume is devoted to learning dynamics and optimization, covering gradient methods, Newton-type methods, stochastic gradient descent, constrained optimization, duality, proximal methods, derivative-free optimization, and support vector machines. The final chapters connect these tools to linear inverse problems, regularized least squares, sparse recovery, dictionary learning, and total least squares.

The book is intended for graduate students, researchers, and practitioners seeking a rigorous but accessible mathematical foundation for modern AI.

Aurelio Uncini is Full Professor of Electrical and Computer Engineering at the Department of Information Engineering, Electronics and Telecommunications (DIET), Sapienza University of Rome, Italy, where he founded and leads the Intelligent Signal Processing and Multimedia (ISPAMM) Laboratory. His research has focused for more than three decades on neural networks and learning systems, from adaptive and complex-valued neural architectures for signal processing to modern deep learning, hypercomplex and quaternion neural networks, representation learning, and generative AI. His work also encompasses adaptive and nonlinear signal processing, optimization methods for machine learning, and neural approaches to audio, speech, image, and biomedical signal processing. He is (co-)author of more than 320 scientific publications and two books, Digital Audio Processing Fundamentals (Springer, 2023) and Fundamentals of Adaptive Signal Processing (Springer, 2015). His current research interests include advanced neural architectures, hypercomplex deep learning, quaternion and wavelet neural networks, multimodal learning, diffusion-based generative models, and AI-driven signal processing. He has served as Program Chair of IJCNN 2025, General Chair of IEEE MLSP 2023, and Program Chair of IEEE MLSP 2016.