Hypercomplex-Valued Neural Networks

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.