Hypercomplex-Valued Neural Networks: Mathematical Aspects and Applications presents a systematic introduction to the mathematical foundations, computational models, and selected applications of neural networks defined over hypercomplex algebras. The book explains how replacing real-valued arithmetic with algebra-valued representations changes data encoding, the structure of neural layers, trainable parameters, and interactions among components of multidimensional data. Beginning with the foundations of finite-dimensional real algebras, the book develops the subject from complex-valued and quaternion-valued models to more general hypercomplex neural architectures. Particular emphasis is placed on the role of algebraic multiplication rules as inductive biases, structured parameter sharing, and the relationship between algebraic structure and neural computation.

Key features include:

• mathematical foundations of hypercomplex algebras relevant to neural-network modelling;

• complex-valued,quaternion-valued, and general hypercomplex-valued neural networks;

• real-valued matrix representations and computational implementation of algebra-valued layers;

• structured coupling of related data components and potential parameter efficiency;

• applications to colour image processing, multichannel signal processing, multimodal learning, and multivariate time-series modelling;

• discussion of methodological limitations, benchmarking, robustness, interpretability, and open research problems.

The book is intended for graduate students, researchers, and practitioners in applied mathematics, machine learning, artificial intelligence, signal processing, and related computational fields who seek a mathematically grounded understanding of hypercomplex-valued neural networks.

Agnieszka Niemczynowicz holds an M.Sc. in Mathematics and a Ph.D. in Physics and has over twenty years of academic research and teaching experience. Her expertise spans applied mathematics, mathematical physics, and artificial intelligence, with a focus on hypercomplex-valued neural networks, tensor methods, and explainable AI. Her research combines mathematical foundations with computational methods and applications in time-series forecasting, signal processing, and image analysis. She is the author and co-author of publications in international peer-reviewed journals and has co-authored and co-edited books on neural networks. She has served as principal investigator on national and international research grants and is a co-recipient of the 2022 Doak Prize awarded by the Journal of Sound and Vibration.

Marcos Eduardo Valle is an associate professor in the Department of Applied Mathematics at the University of Campinas (UNICAMP), Brazil. His research focuses on hypercomplex-valued neural networks and lattice computing, emphasizing their mathematical foundations, computational implementations, and applications. Valle develops machine learning models based on geometric and algebraic frameworks, particularly for multichannel signal and image processing. As an IEEE Senior Member, Valle has authored over 60 publications and coordinated numerous research projects. His interdisciplinary research advances computational intelligence across diverse applications, combining concepts and techniques to address both theoretical and practical challenges in computer vision and neural computation.