This book presents a geometric and computational foundation for modern machine learning, centered on the concepts of affine transformation and manifold training. It continues a structured progression from the author's earlier volumes, which introduced machine learning fundamentals and the preparation of datasets for supervised learning, and shifts the focus toward the internal mechanisms by which learning models are constructed.
Affine transformations form the backbone of all parameterized machine learning models, including neural networks. Each affine transformation combines a linear mapping with a translation, preserving essential geometric properties such as collinearity, parallelism, and ratios along straight lines. These affinity-preserving properties provide the mathematical stability required for reliable function approximation and prediction within a feature space.
The book begins by developing the mathematical formulation of affine transformations and examining their geometric interpretation, properties, and behavior. It then introduces the Affine Transformation Unit (ATU) as a fundamental computational building block. By stacking ATUs into Affine Transformation Arrays (ATAs), models can flexibly increase or reduce dimensionality within affine subspaces. When combined with nonlinear activation functions, these structures naturally give rise to multilayer perceptrons (MLPs), a class of models known for their universal approximation capability.
From a geometric learning perspective, supervised machine learning can be understood as a manifold alignment problem, in which the objective of learning is to construct a model manifold that aligns with the underlying data structure. The second half of the book develops this viewpoint, showing how networks of affine transformations, together with nonlinear activation functions acting as a "hinge" or "glue", create complex global manifolds through folding and unfolding processes.
Throughout the book, theory is closely integrated with practical implementation in Python. Mathematical concepts are reinforced through computational examples, making the material suitable for readers seeking both conceptual clarity and hands-on understanding. This book is intended for students, researchers, and practitioners who wish to develop a deeper geometric intuition for machine learning models and neural networks.
