Diffusion Models & Score-Based Generative Modeling: Theorems, Proofs, and Python Implementations (Computational Mathematics Library)

The complete graduate-level reference for diffusion probabilistic modeling, score estimation, and reverse-time SDEs for generative AI.Mathematical foundations. From measure-theoretic probability and stochastic calculus to Fokker-Planck and Kolmogorov equations, martingales, Girsanov change of measure, and Doob transforms, with careful statements of existence, uniqueness, and regularity.Unifying the score perspective. Fisher divergence, de Bruijn identity, Stein operators, and Hyvarinen score matching, with denoising and noise-conditional formulations that bridge to practical training at multiple noise scales.Diffusion constructions. Forward SDE families (VE, VP, sub-VP), reverse-time SDEs, probability flow ODEs, and discrete-time DDPM with a principled ELBO, all derived end-to-end.Likelihoods and diagnostics. Instantaneous change-of-variables, Hutchinson trace estimation, path-space KL, and conditions for likelihood-consistent sampling and ODE debiasing.Numerical analysis for fast, stable samplers. Weak and strong error theory for Euler-Maruyama, Heun, and Milstein; stiffness and log-SNR schedules; high-order ODE methods and predictor-corrector strategies with rigorous error control.Conditioning and inverse problems. Classifier and classifier-free guidance as controlled diffusions, linear inverse problems via diffusion posterior sampling, and measurement-consistent samplers grounded in Bayesian and control-theoretic principles.Beyond Euclidean spaces. Discrete-state jump processes, scores on manifolds and Lie groups, Schrödinger bridges and entropic optimal transport, KL control and HJB equations, latent diffusion, consistency models, and rectified flows, with proofs and practical algorithms. Read more