Chapter 12 · Book pp. 283–296 · August 9, 2026 edition
Diffusion Generative Models
Connect sampling analysis to score-based reverse diffusions and generative modeling.
Begin with 12.1 Open this chapter in the canonical August 9 source ↗Chapter route
This chapter develops forward diffusion, score, time reversal, denoising. Its main destination is to connect the definitions below to the results that later chapters consume.
Core definitions
- The forward diffusion progressively smooths the data law into a tractable reference law.
- The score is the spatial gradient of a chosen log-density representative.
- The reverse-time SDE uses the marginal score to transport noise back toward data.
- Score matching rewrites inaccessible score regression as an estimable objective.
Main results
- Time reversal identifies the exact reverse drift under regular marginal densities.
- Implicit and denoising score-matching objectives recover the score in population form.
- Generation error decomposes into initialization, score approximation, and time-discretization terms.
- Discretization analysis converts integrated score error into terminal distribution error under explicit law and regularity assumptions.
Contents
Why is this chapter route valid?
Analytic contracts
- A score is defined only with respect to a chosen density and representative.
- Reverse-time formulas require time-marginal regularity and a precise filtration statement.
- An L2 score approximation under one law cannot silently be used under another law.
Open boundaries
- Time-reversal theorem
- Score-field representative and integrability
- End-to-end reverse-discretization bound
View Lean formalization
These mappings are evidence links, not a claim that the entire chapter is formalized.
The reverse-time drift uses the score of the forward marginal, and approximation errors become drift errors.