01 Chapter / 8
Probability Objectives, Mixtures and Noise Paths
If a random generator draws four clusters, how can we tell whether it learned their internal distributions rather than four centers?
Learning Objectives
- Define 2D generation target, density and sampling protocol
- Convergence between the torque of the mixed distribution and the separate linear bridge
- Design complementary indicators using pattern collapse and rectangular matching reverses
02 Chapter / 8
DDPM Forward Chains, Cumulative Noise and Training Targets
After repeated Gaussian noise additions, how can we directly express the sample distribution and training target at any step?
Learning Objectives
- Progressively expand and summarize any time conditions distribution
- Distinguishing hyphenic noise, data edges and accumulated noise
- Fixed noise prediction targets and their information boundaries
03 Chapter / 8
DDPM Posteriors, Variational Objectives and Reverse Sampling
Given a noisy sample and a clean endpoint, how do we compute the previous-step Gaussian posterior and connect it to training?
Learning Objectives
- Complete a square to guide Gaussian conditions.
- Rebuilding items and internal KLs from the lower side of Jensen.
- Motivate noise MSE weights and identify label leaks
04 Chapter / 8
Scores, Denoising Identities and Probability-flow ODEs
Why does the same density evolution use a score coefficient of one-half in the probability-flow ODE and one in the reverse SDE?
Learning Objectives
- The following is the average of score's after-effects versus average noise.
- Density Evolution From Test Function Expanding
- Check the direction and coefficient of the reverse SDE versus the probabilities streamODE
05 Chapter / 8
Conditional Flow Matching and Marginal Velocity Fields
Each conditional straight path has its own velocity. Why can regression on those targets recover the marginal velocity field?
Learning Objectives
- Prove the link between return conditions and the projection of a return to marginal speed
- Consistency equation by test function
- Independent calculations of Gaussian mixed teachers and maintenance of sampling information boundaries
06 Chapter / 8
Euler, Heun and Separable Sampling Errors
If coarse steps make a correct velocity field generate poor samples, how do we separate network error from integration error?
Learning Objectives
- To guide the sampling and error scales from the integral
- Recall solver by field against fixed start value
- Learning and numeric problems separated by a rough step from the resolution field
07 Chapter / 8
Train a 2D Velocity Network and Verify Numerical Gradients
When regressing random endpoint differences, how do we check that every network parameter learns and explain a nonzero loss floor?
Learning Objectives
- Accounting network input, parameters and MSE gradient
- Complete full parameter training and explain non-conditional noise
- Read true training evidence and distinguish preservation from recovery capacity
08 Chapter / 8
Statistical Validation, Deliberate Failures and Joint Verification
When one frozen model sometimes looks good, how can multiple seeds and complementary metrics expose distributional failures?
Learning Objectives
- Define complementary distribution indicators and test their blind spots
- Read the real results of the same seed-by-seed freeze model
- Submission of recoverable evidence and accurate statement of unrealized capacity
Diffusion Models and Flow Matching
Build a generative process from noise and velocity fields.
Advanced · 8 lessons · Approx. 40 hours
Scope and Experimental Limits
Eight lessons connect probability paths, diffusion posteriors, score functions, conditional flow matching, and ODE sampling through independent derivations, worked examples, and fully solved exercises. A self-contained CPU project trains every parameter of a neural velocity field for a two-dimensional four-cluster mixture. Analytic teachers, coarse-step collapse, and distribution errors are retained. This is a two-dimensional principles experiment, not an image generator.
Derive DDPM, scores, and Flow Matching from probability and noise. Train a two-dimensional velocity network, then use numerical integration and statistical tests to expose generation failures.
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Course Details
Course Contents
01 / Probability Objectives, Mixtures and Noise Paths02 / DDPM Forward Chains, Cumulative Noise and Training Targets
03 / DDPM Posteriors, Variational Objectives and Reverse Sampling
04 / Scores, Denoising Identities and Probability-flow ODEs
05 / Conditional Flow Matching and Marginal Velocity Fields
All 8 lessons ↗
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