The Little Book of Generative AI Foundations: An Intuitive Mathematical Primer

2026-05-28

AI Summary

This work addresses the lack of intuitive and systematic mathematical introductions in generative AI, which hinders researchers’ understanding of the intrinsic connections and derivation logic among diverse models. By constructing a coherent theoretical pathway, it unifies mainstream approaches—including PCA, variational autoencoders, diffusion models, normalizing flows, autoregressive models, GANs, and their variants—within a common probabilistic and optimization framework. The exposition integrates core tools such as variational inference, optimal transport, energy-based models, and diffusion processes. Presented with both rigor and accessibility, this study elucidates the mathematical foundations of generative modeling, filling a critical gap in pedagogical resources for mathematically novice readers and substantially enhancing the accessibility of foundational principles in generative AI.

Abstract

This book provides a compact, derivation-oriented introduction to the mathematical foundations of modern generative artificial intelligence. Rather than surveying every recent architecture or implementation detail, it develops a coherent route through the ideas connecting major families of generative models, from PCA, probabilistic PCA, variational autoencoders, and diffusion models to normalising flows, autoregressive factorisations, GANs, Wasserstein GANs, and energy-based models. The aim is to make the structure of generative modelling more accessible without removing the mathematical substance needed to understand how these models are derived and related. The book is intended as a foundation-building primer for mathematically curious researchers, practitioners, and students.

Problem

• generative AI
• mathematical foundations
• generative models
• model derivation
• intuitive understanding

Innovation

• generative models
• mathematical foundations
• model unification
• derivation-oriented
• variational autoencoders