Machine Learning: Affine Transformation and Manifold with Python

Gui-Rong Liu

Author

Dr. G.R. (Guirong) Liu is an expert in computational mechanics, particularly known for pioneering work in meshfree methods, smoothed finite element methods (S-FEM), and particle-based simulations. He currently serves as a Professor of Aerospace Engineering and Engineering Mechanics at the University of Cincinnati. A prolific researcher and educator, Dr. Liu is recognized for pioneering meshfree methods, smoothed finite element methods (S-FEM), and smoothed particle hydrodynamics (SPH). These innovations have significantly advanced simulation techniques in both solid and fluid mechanics. In recent years, Dr. Liu has authored several textbooks and reference works for courses and research in areas such as artificial intelligence, machine learning, mathematics, computational methods, mechanics of materials, solid mechanics, engineering mechanics, applied mechanics, and fluid dynamics.

Synopsis

This book presents a geometric and computational foundation for modern machine learning, centeredon the concepts of affine transformation and manifold training. It continues a structuredprogression from the author’s earlier volumes, which introduced machine learning fundamentalsand the preparation of datasets for supervised learning, and shifts the focus toward the internalmechanisms by which learning models are constructed.Affine transformations form the backbone of all parameterized machine learning models, includingneural networks. Each affine transformation combines a linear mapping with a translation, preservingessential geometric properties such as collinearity, parallelism, and ratios along straight lines.These affinity-preserving properties provide the mathematical stability required for reliable functionapproximation and prediction within a feature space.The book begins by developing the mathematical formulation of affine transformations and examiningtheir geometric interpretation, properties, and behavior. It then introduces the Affine TransformationUnit (ATU) as a fundamental computational building block. By stacking ATUs intoAffine Transformation Arrays (ATAs), models can flexibly increase or reduce dimensionalitywithin affine subspaces. When combined with nonlinear activation functions, these structures naturallygive rise to multilayer perceptrons (MLPs), a class of models known for their universalapproximation capability.From a geometric learning perspective, supervised machine learning can be understood as a manifoldalignment problem, in which the objective of learning is to construct a model manifold thataligns 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 actingas 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. Mathematicalconcepts are reinforced through computational examples, making the material suitable forreaders 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 machinelearning models and neural networks.