Diffusion Model Sampling and Inference covers the techniques for generating high-quality samples from trained diffusion models — including DDPM's stochastic sampling, DDIM's deterministic fast sampling, classifier-free guidance for controllable generation, and advanced schedulers (DPM-Solver, Euler) that reduce the number of denoising steps from 1000 to as few as 1-4 while maintaining quality.

The Diffusion Process

Forward (noising):  x₀ →  x₁ → ... → x_T ≈ N(0,I)
  q(x_t | x_{t-1}) = N(x_t; √(1-β_t)·x_{t-1}, β_t·I)

Reverse (denoising): x_T → x_{T-1} → ... → x₀ (generated image)
  p_θ(x_{t-1} | x_t) = N(x_{t-1}; μ_θ(x_t, t), σ²_t·I)
  The neural network predicts ε_θ(x_t, t) — the noise to remove

DDPM (Denoising Diffusion Probabilistic Models)

Original sampling: iterate T=1000 steps, each adding a small amount of Gaussian noise:

# DDPM sampling (stochastic)
x = torch.randn(shape)  # Start from pure noise
for t in reversed(range(T)):  # T=1000 steps
    predicted_noise = model(x, t)
    x = (1/√α_t) * (x - (β_t/√(1-ᾱ_t)) * predicted_noise)
    if t > 0:
        x += σ_t * torch.randn_like(x)  # stochastic noise

Slow: 1000 forward passes through the U-Net for one image.

DDIM (Denoising Diffusion Implicit Models)

Key insight: derive a deterministic sampling process that skips steps:

# DDIM: deterministic, can use S << T steps (e.g., S=50)
for i, t in enumerate(reversed(subsequence)):  # S=50 steps
    pred_noise = model(x, t)
    pred_x0 = (x - √(1-ᾱ_t) * pred_noise) / √ᾱ_t
    x = √ᾱ_{t-1} * pred_x0 + √(1-ᾱ_{t-1}) * pred_noise
    # No random noise! Deterministic mapping from x_T → x_0

Benefits: 20× fewer steps (50 vs 1000), deterministic (same noise → same image), enables interpolation in latent space.

Classifier-Free Guidance (CFG)

The most impactful technique for controllable generation:

# During training: randomly drop conditioning c with probability p_drop
# During inference: combine conditional and unconditional predictions
pred_uncond = model(x_t, t, null_condition)  # unconditional
pred_cond = model(x_t, t, condition)         # conditional (text prompt)
pred = pred_uncond + w * (pred_cond - pred_uncond)  # w = guidance scale
# w=1: no guidance, w=7.5: typical for Stable Diffusion, w>10: strong guidance

Higher guidance scale → images more closely match the text prompt but with less diversity and potential artifacts. CFG essentially amplifies the signal from the conditioning.

Advanced Samplers

Noise Schedules

The sequence of noise levels β₁...β_T significantly affects quality:

• Linear: β linearly from 10⁻⁴ to 0.02 (original DDPM)
• Cosine: smoother transition, better for small images
• Scaled linear: used in Stable Diffusion, shifted for latent space

Diffusion sampling optimization has been the key enabler of practical generative AI — reducing generation from minutes (1000-step DDPM) to sub-second (1-4 step distilled models) while maintaining the remarkable quality and controllability that made diffusion models the dominant paradigm for image and video generation.