The Principles of Diffusion Models (over 400 pages) (arxiv.org)

A comprehensive monograph titled "The Principles of Diffusion Models" (over 400 pages) lays out a unified mathematical foundation for diffusion-based generative modeling, tracing how diverse formulations arise from shared principles. Rather than proposing a single new algorithm, the work synthesizes three complementary perspectives—variational (VAE-inspired denoising), score-based (energy-model gradients of the evolving density), and flow-based (velocity-field-driven mappings related to normalizing flows)—and shows they all rest on a common time-dependent velocity field whose flow transports a simple prior into the data distribution. The book also surveys practical topics such as controllable generation via guidance, efficient numerical solvers for sampling, and diffusion-motivated flow-map models that learn direct mappings between arbitrary times, with code and demos available. Technically, the monograph frames the forward process as gradually corrupting data into noise and the learning objective as recovering a reverse process that solves an SDE/ODE to evolve noise back into data along a continuous trajectory. Key implications for the AI/ML community include clearer connections between SDE/ODE solvers and sample quality/efficiency, principled approaches to guidance and conditional generation, and new perspectives for designing architectures that approximate velocity or score fields. It’s a rigorous, accessible resource for researchers and practitioners seeking conceptual clarity and practical guidance in diffusion model design and deployment.