Diffusion Models and Their Applications

U1 Diffusion Models 0 / 6 From "what does generation learn" to the forward process, denoising regression, Tweedie & score, and DDPM / DDIM / SDE-ODE sampling.

Tools this unit uses

L0.0 Learning Rules from Examples L0.1 A Loss Function Declares What Matters L0.3 Direction, Step Size, and Gradient Descent L1.0 Overfitting and Underfitting L1.1 Bias and Variance L1.2 Why Split Data? L2.0 Minibatch Gradients and Noise L2.1 Momentum and Adam L4.0 If MLPs Can Approximate Everything, Why Use Other Architectures? L4.5 Where should instructions enter a shared model? L5.0 How should we score how well a model explains the data? L5.1 What can we optimize when the required marginal is intractable? L5.3 Which number tells us whether a generative model is good? M0.3 Total Derivatives, Integral Differentiation, and JVPs M1.0 Linear Combinations of Gaussians M1.1 Conditional Expectation M1.2 Which Temperature Should We Forecast? MSE Selects the Conditional Mean M1.3 Bayes' Rule and the Posterior M1.4 Tweedie's Formula M2.2 The Continuity Equation M2.3 Euler's Method and Its Error M3.0 Where Is the Walker After an Hour? From Random Walks to Brownian Motion M3.1 Stochastic Differential Equations M3.2 The Fokker–Planck Equation M3.3 Langevin Dynamics and Stationarity M4.0 Markov Chains and Transition Matrices M5.1 KL Divergence and Cross-Entropy

U2 Flow Matching 0 / 6 Start from reference paths and explain velocity regression through both theory and intuition. Introduce the continuity equation and its intuitive meaning, use DDIM to bridge Flow Matching and diffusion models, then generalize to stochastic interpolants and a family of ODE/SDE samplers that share the same marginals.

Tools this unit uses

M0.1 Divergence and Flux M1.0 Linear Combinations of Gaussians M1.1 Conditional Expectation M1.2 Which Temperature Should We Forecast? MSE Selects the Conditional Mean M1.4 Tweedie's Formula M2.0 Vector Fields and ODEs M2.2 The Continuity Equation M2.3 Euler's Method and Its Error M5.1 KL Divergence and Cross-Entropy M6.1 How Far Apart Are Two Population Maps? The W₂ Distance

U3 Reducing ODE Discretization Error 0 / 6 Where does ODE discretization error come from? Relate velocity variation to finite-step error, then follow two routes: change learned trajectories with reflow or minibatch OT, or use a better numerical solver for a more accurate approximation under the same compute budget.

Tools this unit uses

M0.2 Taylor Expansion and Straight-Line Prediction M1.1 Conditional Expectation M2.0 Vector Fields and ODEs M2.3 Euler's Method and Its Error M2.4 Higher-Order Methods M3.4 Strong and Weak Convergence M5.0 Convexity and Jensen's Inequality M6.0 Monge, Kantorovich, and Couplings M6.1 How Far Apart Are Two Population Maps? The W₂ Distance M6.2 One-Dimensional Monotonicity and Brenier's Theorem M6.3 Sinkhorn and Minibatch Approximations

U4 Discrete Diffusion 0 / 8 Start from joint PMFs and probability interpolation, then ask what velocity means for discrete data. Jump rates, conservation, and posterior averaging lead to training and sampling; factorization and remasking motivate two labs separating sampling and prediction errors.

Tools this unit uses

M1.1 Conditional Expectation M1.3 Bayes' Rule and the Posterior M2.2 The Continuity Equation M3.2 The Fokker–Planck Equation M3.3 Langevin Dynamics and Stationarity M4.0 Markov Chains and Transition Matrices M4.1 Absorbing States and Absorption Times M4.2 Continuous-Time Markov Chains M4.3 Can You Detect a Reversed Movie? Time Reversal and Probability Ratios M5.1 KL Divergence and Cross-Entropy

U5 Consistency Models 0 / 7 Can we learn the one-step map directly? Progressive distillation, the consistency function and self-consistency, CD vs. CT, which error each iCT / sCM trick fights, and why multistep CM saturates.

Tools this unit uses

M0.2 Taylor Expansion and Straight-Line Prediction M0.3 Total Derivatives, Integral Differentiation, and JVPs M1.1 Conditional Expectation M1.2 Which Temperature Should We Forecast? MSE Selects the Conditional Mean M1.4 Tweedie's Formula M2.0 Vector Fields and ODEs M2.1 Flow Maps and Pushforwards M2.3 Euler's Method and Its Error M5.2 EMA, Moving Targets, and Stop-Gradient M6.1 How Far Apart Are Two Population Maps? The W₂ Distance

U6 Consistency Trajectory Model and Flow Map 0 / 7 Start from CTM's arbitrary t→s traversal, then organize it as a flow map: four defining conditions, the three Flow Map Matching losses, the MeanFlow identity, Shortcut / AYF, and failure modes of regression versus distribution-matching (DMD) distillation.

Tools this unit uses

M0.3 Total Derivatives, Integral Differentiation, and JVPs M1.1 Conditional Expectation M1.2 Which Temperature Should We Forecast? MSE Selects the Conditional Mean M1.4 Tweedie's Formula M2.0 Vector Fields and ODEs M2.1 Flow Maps and Pushforwards M2.3 Euler's Method and Its Error M5.1 KL Divergence and Cross-Entropy M5.3 Differentiating a Generator M6.1 How Far Apart Are Two Population Maps? The W₂ Distance

U7 Test-time Guidance 0 / 6 How can a trained model incorporate a new condition, observation, or reward at sampling time? From classifier guidance, CFG, and reward tilting to Best-of-N, Feynman–Kac/SMC, SPT, inverse problems, data assimilation, planning, and scientific design.

Tools this unit uses

M1.3 Bayes' Rule and the Posterior M3.3 Langevin Dynamics and Stationarity M4.0 Markov Chains and Transition Matrices M5.1 KL Divergence and Cross-Entropy