ChungMG

Diffusion Model

Diffusion Models are a class of generative models that simulate the process of data being gradually corrupted by noise (forward diffusion), and then learn the reverse process to recover data from noise.

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1. Core Idea

Diffusion models consist of two core processes:

Process	Direction	Description
Forward Diffusion	Data $\to$ Noise	Gradually add Gaussian noise to data until it becomes pure noise
Reverse Denoising	Noise $\to$ Data	Learn to gradually recover original data from noise

[!NOTE] Physical Analogy
Similar to diffusion phenomenon in thermodynamics: a drop of ink in water gradually diffuses until uniformly distributed. The reverse process is “condensing” back to the initial state from uniform distribution.

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2. Forward Diffusion Process

2.1 Discrete-Time Formulation (DDPM)

Given data point $x_0\sim q(x)$ , gradually add Gaussian noise:

$$q(x_t\mid x_{t-1})=\mathcal{N}(x_t;\sqrt{1-{\beta }_t}{x}_{t-1},{\beta }_{t}I)$$

where ${\beta }_t\in (0,1)$ is the noise schedule, typically satisfying ${\beta }_1<{\beta }_2<\cdots <{\beta }_T$ .

After $T$ steps, $x_T$ approximates standard Gaussian distribution $\mathcal{N}(0,I)$ .

Common Noise Schedules:

Schedule	Formula	Characteristics
Linear	${\beta }_t={\beta }_1+(t-1)\frac{{\beta }_T-{\beta }_1}{T-1}$	Simple, widely used
Cosine	${\overline{\alpha }}_t=\frac{f(t)}{f(0)}$ , $f(t)=\cos {(\frac{t/T+s}{1+s}\cdot \frac{\pi }{2})}^2$	Better for small $t$
Quadratic	${\beta }_t={(\sqrt{{\beta }_1}+(t-1)\frac{\sqrt{{\beta }_T}-\sqrt{{\beta }_1}}{T-1})}^2$	Slower initial noise

2.2 Reparameterization Trick

Define ${\alpha }_t=1-{\beta }_t$ , ${\overline{\alpha }}_t=\prod _{s=1}^t{\alpha }_s$ , then:

$$x_t=\sqrt{{\overline{\alpha }}_t}{x}_0+\sqrt{1-{\overline{\alpha }}_t}ϵ,ϵ\sim \mathcal{N}(0,I)$$

Key Property: We can sample $x_t$ at any timestep directly without iterating:

$$q(x_t\mid x_0)=\mathcal{N}(x_t;\sqrt{{\overline{\alpha }}_t}{x}_0,(1-{\overline{\alpha }}_t)I)$$

2.3 Continuous-Time Formulation ([[Stochastic Differential Equation (SDE)|SDE]])

The forward process can be written as [[Stochastic Differential Equation (SDE)|Stochastic Differential Equation]]:

$$\mathrm{d}x=f(t)x\mathrm{d}t+g(t)\mathrm{d}{W}_t$$

where $W_t$ is [[Wiener Process|Wiener Process]], $f(t)$ is the drift coefficient, and $g(t)$ is the diffusion coefficient.

Discrete-Continuous Correspondence:

$${\overline{\alpha }}_t=\exp (-{\int }_0^t\beta (s)\mathrm{d}s)$$

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3. Reverse Denoising Process

3.1 Discrete-Time Formulation

The reverse process is also modeled as Gaussian distribution:

$$p_{\theta }(x_{t-1}\mid x_t)=\mathcal{N}(x_{t-1};{\mu }_{\theta }(x_t,t),{\mathrm{\Sigma }}_{\theta }(x_t,t))$$

Learn mean ${\mu }_{\theta }$ and variance ${\mathrm{\Sigma }}_{\theta }$ through neural network.

Optimal Reverse Distribution (when ${\beta }_t\to 0$ ):

$$q(x_{t-1}\mid x_t,x_0)=\mathcal{N}(x_{t-1};\frac{\sqrt{{\overline{\alpha }}_{t-1}}{\beta }_t}{1-{\overline{\alpha }}_t}{x}_0+\frac{\sqrt{{\alpha }_t}(1-{\overline{\alpha }}_{t-1})}{1-{\overline{\alpha }}_t}{x}_t,\frac{1-{\overline{\alpha }}_{t-1}}{1-{\overline{\alpha }}_t}{\beta }_{t}I)$$

3.2 Simplified Training Objective (DDPM)

Ho et al. (2020) proposed simplified loss function:

$$L_{\text{simple}}={\mathbb{E}}_{t,x_0,ϵ}[\parallel ϵ-{ϵ}_{\theta }(x_t,t){\parallel }^2]$$

• $ϵ\sim \mathcal{N}(0,I)$ : true noise
• ${ϵ}_{\theta }(x_t,t)$ : noise predicted by neural network

Full Variational Lower Bound:

$$L_{\text{VLB}}={\mathbb{E}}_q[D_{\text{KL}}(q(x_T|x_0)\parallel p(x_T))+\sum _{t=2}^T{D}_{\text{KL}}(q(x_{t-1}|x_t,x_0)\parallel p_{\theta }(x_{t-1}|x_t))-\log p_{\theta }(x_0|x_1)]$$

3.3 Continuous-Time Formulation (Score-based)

Reverse [[Stochastic Differential Equation (SDE)|SDE]]:

$$\mathrm{d}x=[f(t)x-g(t{)}^2{\mathrm{\nabla }}_x\log p_t(x)]\mathrm{d}t+g(t)\mathrm{d}{\overline{W}}_t$$

where ${\overline{W}}_t$ is [[Wiener Process|Wiener Process]] in reverse time, and ${\mathrm{\nabla }}_x\log p_t(x)$ is [[Score Function|Score Function]].

[[Score Function]] Estimation:

The [[Score Function]] is learned via score matching:

$$\mathcal{L}(\theta )=\frac{1}{2}{\mathbb{E}}_t{\mathbb{E}}_{x_0}{\mathbb{E}}_{x_t|x_0}[\parallel s_{\theta }(x_t,t)-{\mathrm{\nabla }}_{x_t}\log q(x_t|x_0){\parallel }^2]$$

where ${\mathrm{\nabla }}_{x_t}\log q(x_t|x_0)=-\frac{x_t-\sqrt{{\overline{\alpha }}_t}{x}_0}{1-{\overline{\alpha }}_t}$ .

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4. Core Formula Summary

[!QUOTE] DDPM Forward Noising

$$x_t=\sqrt{{\overline{\alpha }}_t}{x}_0+\sqrt{1-{\overline{\alpha }}_t}ϵ,ϵ\sim \mathcal{N}(0,I)$$

[!QUOTE] DDPM Simplified Loss

$$L_{\text{simple}}={\mathbb{E}}_{t,x_0,ϵ}[\parallel ϵ-{ϵ}_{\theta }(x_t,t){\parallel }^2]$$

[!QUOTE] Forward [[Stochastic Differential Equation (SDE)|SDE]]

$$\mathrm{d}x=f(t)x\mathrm{d}t+g(t)\mathrm{d}{W}_t$$

[!QUOTE] Reverse [[Stochastic Differential Equation (SDE)|SDE]]

$$\mathrm{d}x=[f(t)x-g(t{)}^2{\mathrm{\nabla }}_x\log p_t(x)]\mathrm{d}t+g(t)\mathrm{d}{\overline{W}}_t$$

[!QUOTE] [[Probability Flow ODE]]

$$\mathrm{d}x=[f(t)x-\frac{1}{2}g(t{)}^2{\mathrm{\nabla }}_x\log p_t(x)]\mathrm{d}t$$

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5. Main Variants

Model	Features	Key Contributions
DDPM	Discrete-time, pixel space	Established the basic framework of diffusion models
DDIM	Deterministic sampling, accelerated generation	Non-Markovian forward process, supports skip-step sampling
Score [[Stochastic Differential Equation (SDE)|SDE]]	Continuous-time [[Stochastic Differential Equation (SDE)|SDE]] framework	Unified DDPM and Score Matching
LDM	Latent space diffusion	Perform diffusion in VAE latent space, reducing computation
DiT	Transformer architecture	Use Transformer instead of U-Net
EDM	Improved design choices	Better architecture, sampling, and training
Stable Diffusion	Text-conditional LDM	Cross-attention for text guidance, widely adopted

5.1 DDIM (Denoising Diffusion Implicit Models)

DDIM generalizes DDPM to non-Markovian processes:

$$x_{t-1}=\sqrt{{\overline{\alpha }}_{t-1}}\left(\frac{x_t-\sqrt{1-{\overline{\alpha }}_t}{ϵ}_{\theta }(x_t,t)}{\sqrt{{\overline{\alpha }}_t}}\right)+\sqrt{1-{\overline{\alpha }}_{t-1}-{\sigma }_t^2}\cdot {ϵ}_{\theta }(x_t,t)+{\sigma }_{t}z$$

where ${\sigma }_t=\eta \sqrt{\frac{1-{\overline{\alpha }}_{t-1}}{1-{\overline{\alpha }}_t}}\sqrt{1-\frac{{\overline{\alpha }}_t}{{\overline{\alpha }}_{t-1}}}$ , and $\eta \in [0,1]$ :

• $\eta =1$ : DDPM (stochastic)
• $\eta =0$ : DDIM (deterministic)

Key Advantage: Can use fewer timesteps (e.g., 50 instead of 1000) for faster sampling.

5.2 LDM (Latent Diffusion Models)

Instead of diffusing in pixel space, LDM operates in latent space:

1. Compress: $z=\mathcal{E}(x)$ using VAE encoder
2. Diffuse: Apply diffusion process to $z$
3. Decode: $\hat{x}=\mathcal{D}(z_0)$ using VAE decoder

Benefits:

• Lower dimensionality (e.g., $64\times 64\times 4$ vs $512\times 512\times 3$ )
• Faster training and inference
• Perceptual compression preserves semantic information

5.3 DiT (Diffusion Transformers)

Replace U-Net with Transformer architecture:

• Patching: Split image into patches (like ViT)
• Self-attention: Capture long-range dependencies
• Scaling: Better performance with larger models
• Flexibility: Easy to incorporate conditioning

Result: DiT-XL/2 outperforms U-Net on ImageNet generation.

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6. Training and Sampling Algorithms

6.1 Training Loop

# Pseudocode
while not converged:
    x_0 = sample_from_dataset()
    t = sample_uniform(1, T)
    epsilon = sample_normal(0, I)
    
    x_t = sqrt(alpha_bar[t]) * x_0 + sqrt(1 - alpha_bar[t]) * epsilon
    loss = MSE(epsilon, epsilon_theta(x_t, t))
    
    loss.backward()
    optimizer.step()

Training Tricks:

• t weighting: Weight loss by $1/\mathbb{E}[\parallel ϵ{\parallel }^2]$ or use uniform weighting
• Architecture: U-Net with attention, group normalization, SiLU activation
• EMA: Exponential moving average of model weights for better sampling
• Dropout: Apply to attention layers for regularization

6.2 Sampling Loop (DDPM)

x_T = sample_normal(0, I)
for t in reversed(range(1, T+1)):
    z = sample_normal(0, I) if t > 1 else 0
    epsilon = epsilon_theta(x_t, t)
    x_{t-1} = 1/sqrt(alpha_t) * (x_t - (1-alpha_t)/sqrt(1-alpha_bar[t]) * epsilon) + sqrt(beta_t) * z

6.3 Advanced Sampling Methods

Method	Steps	Approach
DDPM	1000	Original stochastic sampling
DDIM	50-100	Deterministic, skip steps
[[DPM-Solver]]	10-20	ODE solver with adaptive steps
[[DPM-Solver]]++	10-15	Improved stability
UniPC	5-10	Unified predictor-corrector
Consistency Models	1-5	Direct mapping, distillation

Predictor-Corrector Framework (for [[Stochastic Differential Equation (SDE)|SDE]]-based models):

1. Predictor: Take one step using reverse [[Stochastic Differential Equation (SDE)|SDE]]/ODE
2. Corrector: Apply Langevin dynamics to refine sample
3. Repeat: Alternate for better quality

# Predictor-Corrector Sampling
for t in reversed(timesteps):
    # Predictor step (Euler-Maruyama)
    score = score_model(x_t, t)
    x_t = x_t + drift(x_t, t) * dt + diffusion(t) * score * dt + noise
    
    # Corrector step (Langevin)
    for _ in range(corrector_steps):
        score = score_model(x_t, t)
        x_t = x_t + step_size * score + sqrt(2 * step_size) * noise

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7. Advantages and Disadvantages

Advantages

• High generation quality: Reaches or exceeds GAN level
• Stable training: No mode collapse problem like GAN
• Elegant theory: Based on thermodynamics and [[Stochastic Differential Equation (SDE)|SDE]] mathematical foundation
• Flexible conditioning: Easy to incorporate text, image, or other conditions
• Coverage: Better mode coverage than GANs (less mode collapse)
• Likelihood estimation: Can compute exact likelihoods (via ODE)

Disadvantages

• Slow sampling speed: Requires tens to hundreds of iterative steps
• Sensitive to hyperparameters: Noise schedule affects generation quality
• High computational cost: Training requires significant resources
• Blurriness: May produce blurry samples compared to GANs (in pixel space)

Acceleration Methods

Algorithm-Level:

• DDIM (Deterministic sampling)
• [[DPM-Solver]] (Ordinary differential equation solver)
• Progressive Distillation
• Consistency Models (One-step generation)

Architecture-Level:

• Latent space diffusion (LDM)
• Distilled models (smaller, faster)
• Quantization and pruning

Hardware-Level:

• GPU optimization
• Parallel sampling
• Mixed precision training

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8. Applications in AI Image Generation

Application	Representative Models	Features
Text-to-Image	DALL-E 2, Stable Diffusion, Imagen	Diffusion model + CLIP + Latent space
Image Editing	InstructPix2Pix, Prompt-to-Prompt	Conditional guided editing
Video Generation	Stable Video Diffusion	Introduce temporal dimension
3D Generation	DreamFusion, Magic3D	Score Distillation Sampling (SDS)
Image Super-Resolution	SR3, RePaint	Diffusion + Denoising
Inpainting	Stable Diffusion, GLIDE	Mask-guided generation
Style Transfer	StyleDrop, Custom Diffusion	Style adaptation
Controlled Generation	ControlNet, T2I-Adapter	Spatial control signals

8.1 Text-to-Image Generation

Architecture:

1. Text Encoder: CLIP, T5, or custom transformer
2. Conditioning: Cross-attention in U-Net/DiT
3. Diffusion: Latent space denoising
4. Decoder: VAE decoder to pixel space

Training Data: LAION-5B, COCO, internal datasets

8.2 Image-to-Image Translation

Given source image $x_{\text{src}}$ and target description:

$$x_{\text{result}}=\text{Sample}(x_T\to x_0\mid x_{\text{src}},\text{text})$$

Methods:

• Img2Img: Add noise to source, then denoise with conditioning
• ControlNet: Copy and adapt U-Net weights for control
• IP-Adapter: Image prompt adapter for visual conditioning

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9. Conditional Diffusion Models

9.1 Classifier Guidance

$${\widetilde{ϵ}}_{\theta }(x_t,t,c)={ϵ}_{\theta }(x_t,t)-\sqrt{1-{\overline{\alpha }}_t}\cdot {\mathrm{\nabla }}_{x_t}\log p_{\varphi }(c\mid x_t)$$

Pros:

• Works with pre-trained classifiers
• Flexible guidance strength

Cons:

• Requires training separate classifier
• Limited to classification conditions

9.2 Classifier-Free Guidance

$${\widetilde{ϵ}}_{\theta }(x_t,t,c)=(1+w)\cdot {ϵ}_{\theta }(x_t,t,c)-w\cdot {ϵ}_{\theta }(x_t,t)$$

where $w>0$ is the guidance strength.

Training: Randomly drop condition (e.g., 10% probability) during training to learn unconditional model.

Pros:

• No separate classifier needed
• Works with any condition type (text, image, etc.)
• Better quality than classifier guidance

Cons:

• Requires larger model (learns conditional + unconditional)
• Guidance strength $w$ needs tuning

9.3 Multi-Modal Conditioning

Modern diffusion models support multiple conditions:

Condition Type	Encoding Method	Integration
Text	CLIP, T5 transformer	Cross-attention
Image	CLIP vision, VAE encoder	Concatenation, attention
Depth/Edges	CNN encoder	ControlNet, adapter
Pose/Skeleton	Graph neural network	Spatial injection
Audio	VGGish, CLAP	Cross-attention

9.4 Controllability Methods

ControlNet:

• Clone U-Net encoder layers
• Train with zero convolution initialization
• Lock original model, train control branches

IP-Adapter:

• Add image encoder parallel to text encoder
• Use decoupled cross-attention
• Enables image prompt guidance

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10. Theoretical Analysis

10.1 Connection to Variational Inference

Diffusion models optimize the variational lower bound (ELBO):

$$\log p_{\theta }(x_0)\ge {\mathbb{E}}_q[\log p_{\theta }(x_0|x_1)]-\sum _{t=2}^T{D}_{\text{KL}}(q(x_{t-1}|x_t,x_0)\parallel p_{\theta }(x_{t-1}|x_t))-D_{\text{KL}}(q(x_T|x_0)\parallel p(x_T))$$

Interpretation:

• Term 1: Reconstruction loss
• Terms 2: Consistency between forward and reverse processes
• Term 3: Prior matching (ensure $x_T$ is close to Gaussian)

10.2 Connection to Score Matching

Score matching objective:

$$\mathcal{J}(\theta )=\frac{1}{2}{\mathbb{E}}_{p(x)}[\parallel s_{\theta }(x)-{\mathrm{\nabla }}_x\log p(x){\parallel }^2]$$

For diffusion models, this becomes denoising score matching:

$$\mathcal{J}(\theta )=\frac{1}{2}\sum _{t=1}^T{\mathbb{E}}_{x_0,x_t}[\parallel s_{\theta }(x_t,t)-{\mathrm{\nabla }}_{x_t}\log q(x_t|x_0){\parallel }^2]$$

10.3 Neural Tangent Kernel (NTK) Analysis

In the infinite-width limit, diffusion model training can be analyzed via NTK:

• Training dynamics: Governed by kernel regression
• Generalization: Related to kernel eigenvalues
• Mode coverage: Depends on data spectrum

10.4 Information Bottleneck Perspective

Forward diffusion as information bottleneck:

$$I(x_0;x_t)=\text{Information preserved at time }t$$

• Early timesteps: High mutual information (preserve details)
• Late timesteps: Low mutual information (only semantic info)
• Optimal schedule balances compression and preservation

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11. Core Formula Cards

[!QUOTE] Reparameterization Noising

$$x_t=\sqrt{{\overline{\alpha }}_t}{x}_0+\sqrt{1-{\overline{\alpha }}_t}ϵ$$

[!QUOTE] DDPM Loss

$$L_{\text{simple}}=\parallel ϵ-{ϵ}_{\theta }(x_t,t){\parallel }^2$$

[!QUOTE] DDIM Sampling

$$x_{t-1}=\sqrt{{\overline{\alpha }}_{t-1}}\left(\frac{x_t-\sqrt{1-{\overline{\alpha }}_t}{ϵ}_{\theta }(x_t,t)}{\sqrt{{\overline{\alpha }}_t}}\right)+\sqrt{1-{\overline{\alpha }}_{t-1}-{\sigma }_t^2}{ϵ}_{\theta }(x_t,t)+{\sigma }_{t}z$$

[!QUOTE] Classifier-Free Guidance

$$\widetilde{ϵ}={ϵ}_{\theta }(x_t,t,\varnothing )+w\cdot ({ϵ}_{\theta }(x_t,t,c)-{ϵ}_{\theta }(x_t,t,\varnothing ))$$

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12. Evaluation Metrics

12.1 Sample Quality

Metric	Description	Range
FID	Fréchet Inception Distance	Lower is better (0 is perfect)
IS	Inception Score	Higher is better
Precision/Recall	Quality vs. diversity trade-off	[0, 1]
KID	Kernel Inception Distance	Lower is better

FID Formula:

$$\text{FID}=\parallel {\mu }_r-{\mu }_g{\parallel }^2+\text{Tr}({\mathrm{\Sigma }}_r+{\mathrm{\Sigma }}_g-2({\mathrm{\Sigma }}_r{\mathrm{\Sigma }}_g{)}^{1/2})$$

where ${\mu }_r,{\mathrm{\Sigma }}_r$ are real data statistics and ${\mu }_g,{\mathrm{\Sigma }}_g$ are generated statistics.

12.2 Likelihood Evaluation

Bits per dimension (bpd):

$$\text{bpd}=-\frac{{\log }_2{p}_{\theta }(x)}{\text{dim}(x)}$$

Lower bpd indicates better likelihood.

12.3 Diversity Metrics

• Mode coverage: Percentage of data modes captured
• LPIPS: Learned Perceptual Image Patch Similarity (diversity)
• Unique samples: Ratio of unique generated samples

12.4 Human Evaluation

• User studies: Preference ratings
• Text-image alignment: CLIP score for text-to-image
• Aesthetic quality: Aesthetic score predictors

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13. Practical Implementation Tips

13.1 Network Architecture

U-Net Design:

Input
  ↓
Downsample Block 1 (128 channels)
  ↓
Downsample Block 2 (256 channels)
  ↓
Downsample Block 3 (512 channels)
  ↓
Middle Block with Attention (1024 channels)
  ↓
Upsample Block 3 (512 channels) + Skip Connection
  ↓
Upsample Block 2 (256 channels) + Skip Connection
  ↓
Upsample Block 1 (128 channels) + Skip Connection
  ↓
Output (3 channels)

Key Components:

• ResNet blocks: Groups of 2-3 conv layers with skip connections
• Attention: Self-attention at lowest resolution (e.g., 32x32)
• Time embedding: Sinusoidal position encoding → MLP
• Conditioning: Cross-attention for text, AdaGN for class labels

13.2 Training Best Practices

Hyperparameters:

Parameter	Recommended Value	Notes
Timesteps	1000	Standard, can use fewer for fast sampling
Batch size	256-512	Larger is better if memory allows
Learning rate	1e-4	Use cosine decay schedule
Optimizer	AdamW	β₁=0.9, β₂=0.999
EMA rate	0.9999	Exponential moving average
Gradient clipping	1.0	Prevents explosion

Data Augmentation:

• Random horizontal flip
• Random crop and resize
• No color jitter (changes data distribution)

13.3 Debugging Checklist

✓ Check noise schedule: Plot ${\overline{\alpha }}_t$ vs $t$ , ensure smooth decay
✓ Monitor loss curves: Should decrease smoothly, no spikes
✓ Validate reparameterization: $x_t$ should match theoretical distribution
✓ Test sampling: Start with small model, verify basic functionality
✓ Check gradients: Norm should be reasonable (< 10)
✓ Visualize intermediates: Sample at different timesteps during training

13.4 Common Issues and Solutions

Problem	Cause	Solution
Blurry samples	Undertraining, high noise	Train longer, check schedule
Mode collapse	Low capacity, overfitting	Increase model size, add dropout
Training instability	High learning rate	Reduce LR, add gradient clipping
Slow sampling	Too many timesteps	Use DDIM, [[DPM-Solver]]
Poor conditioning	Weak guidance	Increase guidance strength $w$

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14. Recent Advances (2023-2024)

14.1 Consistency Models

Key Idea: Learn direct mapping from noise to data in one step.

$$f_{\theta }(x_t,t)\approx x_0\mathrm{\forall }t$$

Benefits:

• 1-step generation (vs. 1000 steps)
• Distillation from pre-trained diffusion models
• Competitive quality with fewer steps

14.2 Rectified Flows

Concept: Learn straight trajectories between noise and data.

$$\frac{\mathrm{d}x}{\mathrm{d}t}=v_{\theta }(x,t)$$

where $v_{\theta }$ is velocity field pointing from $x_1$ (noise) to $x_0$ (data).

Advantage: Fewer integration steps needed.

14.3 Diffusion Transformers (DiT)

• Replace U-Net with Transformer
• Scale to billions of parameters
• Better performance with larger models
• Used in SORA, Stable Diffusion 3

14.4 [[Flow Matching]]

General framework encompassing diffusion models:

$$\frac{\mathrm{d}x}{\mathrm{d}t}=u_t(x)$$

where $u_t(x)$ is learned vector field.

Unifies:

• Diffusion models
• Normalizing flows
• Continuous normalizing flows

14.5 Video Diffusion Models

Challenges:

• Temporal consistency
• Computational cost (3D + time)
• Long sequence generation

Solutions:

• Spatiotemporal attention
• Cascaded generation
• Latent video diffusion

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15. Comparison with Other Generative Models

Model	Quality	Diversity	Training Stability	Sampling Speed	Likelihood
GAN	★★★★☆	★★☆☆☆	★☆☆☆☆	★★★★★	✗
VAE	★★☆☆☆	★★★★☆	★★★★★	★★★★★	✓
Diffusion	★★★★★	★★★★★	★★★★★	★★☆☆☆	✓
Flow	★★★☆☆	★★★★☆	★★★★☆	★★★☆☆	✓
EBM	★★★★☆	★★★☆☆	★★☆☆☆	★☆☆☆☆	✓

When to use Diffusion Models:

• ✓ Need high-quality samples
• ✓ Mode coverage is important
• ✓ Training stability is critical
• ✗ Real-time generation required
• ✗ Limited computational resources

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Related Concepts

• [[Wiener Process|Wiener Process]]
• [[Stochastic Differential Equation (SDE)|SDE]]
• [[Score Function]]
• [[Probability Flow ODE]]
• [[Fokker-Planck Equation]]
• [[Kolmogorov Equations]]
• [[DDIM]]
• [[DPM-Solver]]
• [[Flow Matching]]
• [[Markov Process]]
• [[Neural ODE]]
• [[ResNet]]
• [[U-Net]]
• [[DiT]]
• [[Vision Transformer (ViT)]]
• [[Variational Autoencoder (VAE)]]
• [[Generative Adversarial Network (GAN)]]
• [[Langevin Dynamics]]
• [[Denoising Score Matching]]
• [[Itô Integral]]
• [[Martingale]]

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References

• Paper：Denoising Diffusion Probabilistic Models (Ho et al., 2020)
• Paper：Score-Based Generative Modeling through SDEs (Song et al., 2021)
• Paper：High-Resolution Image Synthesis with Latent Diffusion Models (Rombach et al., 2022)
• Blog：What are Diffusion Models? - Lilian Weng
• Course：CS236 Deep Generative Models (Stanford)