Volume V
Diffusion Models
Forward and reverse diffusion processes, score matching, and conditional generation.
Mathematical Foundations
Forward Diffusion Process
Volume III, Chapter 10 — Part I. Rigorous derivation of the forward diffusion Markov chain, closed-form marginal via induction, noise schedules, signal-to-noise ratio, and connection to the ε-prediction training objective.
Reverse Process & Denoising
Volume III, Chapter 10 — Part II. Tractable posterior q(x_{t-1} given x_t and x_0), variational lower bound, ε-prediction parameterization, score matching connection, and DDPM sampling theory.
Score Matching & Score-Based Models
The score function, denoising score matching, Stein's identity, connection between score-based and diffusion models, Langevin dynamics sampling, and noise-conditional score networks.
Training & Sampling
DDPM Training & the Noise Prediction Objective
Complete derivation of the DDPM training loss: variational lower bound, simplification to noise prediction, loss weighting strategies, training algorithm, and connection to score matching.
DDIM & Accelerated Sampling
Denoising Diffusion Implicit Models: deterministic sampling, the DDIM update rule, connection to probability flow ODE, DPM-Solver, consistency models, and reducing sampling from 1000 to 4 steps.
Noise Schedules & Signal-to-Noise Ratio
Linear, cosine, and learned noise schedules: derivation, SNR analysis, the effect of schedule choice on sample quality, and continuous-time formulations via SDEs.
Architectures
The U-Net Architecture for Diffusion
The denoising backbone: encoder-decoder with skip connections, time embedding injection, self-attention at multiple scales, cross-attention for conditioning, and the DiT (Diffusion Transformer) alternative.
Latent Diffusion Models (Stable Diffusion)
Diffusion in compressed latent space: VAE encoder/decoder, the latent space advantage, training pipeline, the full Stable Diffusion architecture, and efficiency analysis.
Conditioning & Guidance
Classifier-Free Guidance
Volume III, Chapter 10 — Part III. Conditioning diffusion models without a classifier: joint training, guided score derivation, guidance scale effects, and the quality-diversity tradeoff.
ControlNet & Spatial Conditioning
Adding spatial control to diffusion: ControlNet architecture (zero-convolution), IP-Adapter, T2I-Adapter, multi-conditioning, and the theory of conditional diffusion with spatial signals.