Diffusion models have emerged as a strong approach in generative AI, producing state-of-the-art leads to image, audio, and video generation. On this in-depth technical article, we’ll explore how diffusion models work, their key innovations, and why they’ve change into so successful. We’ll cover the mathematical foundations, training process, sampling algorithms, and cutting-edge applications of this exciting recent technology.
Introduction to Diffusion Models
Diffusion models are a category of generative models that learn to regularly denoise data by reversing a diffusion process. The core idea is to begin with pure noise and iteratively refine it right into a high-quality sample from the goal distribution.
This approach was inspired by non-equilibrium thermodynamics – specifically, the strategy of reversing diffusion to get better structure. Within the context of machine learning, we are able to consider it as learning to reverse the gradual addition of noise to data.
Some key benefits of diffusion models include:
- State-of-the-art image quality, surpassing GANs in lots of cases
- Stable training without adversarial dynamics
- Highly parallelizable
- Flexible architecture – any model that maps inputs to outputs of the identical dimensionality could be used
- Strong theoretical grounding
Let’s dive deeper into how diffusion models work.
Source: Song et al.
Stochastic Differential Equations govern the forward and reverse processes in diffusion models. The forward SDE adds noise to the information, regularly transforming it right into a noise distribution. The reverse SDE, guided by a learned rating function, progressively removes noise, resulting in the generation of realistic images from random noise. This approach is vital to achieving high-quality generative performance in continuous state spaces
The Forward Diffusion Process
The forward diffusion process starts with an information point x₀ sampled from the true data distribution, and regularly adds Gaussian noise over T timesteps to supply increasingly noisy versions x₁, x₂, …, xT.
At each timestep t, we add a small amount of noise in accordance with:
x_t = √(1 - β_t) * x_{t-1} + √(β_t) * ε
Where:
- β_t is a variance schedule that controls how much noise is added at each step
- ε is random Gaussian noise
This process continues until xT is almost pure Gaussian noise.
Mathematically, we are able to describe this as a Markov chain:
q(x_t | x_{t-1}) = N(x_t; √(1 - β_t) * x_{t-1}, β_t * I)
Where N denotes a Gaussian distribution.
The β_t schedule is usually chosen to be small for early timesteps and increase over time. Common selections include linear, cosine, or sigmoid schedules.
The Reverse Diffusion Process
The goal of a diffusion model is to learn the reverse of this process – to begin with pure noise xT and progressively denoise it to get better a clean sample x₀.
We model this reverse process as:
p_θ(x_{t-1} | x_t) = N(x_{t-1}; μ_θ(x_t, t), σ_θ^2(x_t, t))
Where μ_θ and σ_θ^2 are learned functions (typically neural networks) parameterized by θ.
The important thing innovation is that we need not explicitly model the complete reverse distribution. As a substitute, we are able to parameterize it by way of the forward process, which we all know.
Specifically, we are able to show that the optimal reverse process mean μ* is:
μ* = 1/√(1 - β_t) * (x_t - β_t/√(1 - α_t) * ε_θ(x_t, t))
Where:
- α_t = 1 – β_t
- ε_θ is a learned noise prediction network
This provides us a straightforward objective – train a neural network ε_θ to predict the noise that was added at each step.
Training Objective
The training objective for diffusion models could be derived from variational inference. After some simplification, we arrive at a straightforward L2 loss:
L = E_t,x₀,ε [ ||ε - ε_θ(x_t, t)||² ]
Where:
- t is sampled uniformly from 1 to T
- x₀ is sampled from the training data
- ε is sampled Gaussian noise
- x_t is constructed by adding noise to x₀ in accordance with the forward process
In other words, we’re training the model to predict the noise that was added at each timestep.
Model Architecture
Source: Ronneberger et al.
The U-Net architecture is central to the denoising step within the diffusion model. It features an encoder-decoder structure with skip connections that help preserve fine-grained details in the course of the reconstruction process. The encoder progressively downsamples the input image while capturing high-level features, and the decoder up-samples the encoded features to reconstruct the image. This architecture is especially effective in tasks requiring precise localization, corresponding to image segmentation.
The noise prediction network ε_θ can use any architecture that maps inputs to outputs of the identical dimensionality. U-Net style architectures are a well-liked alternative, especially for image generation tasks.
A typical architecture might appear like:
class DiffusionUNet(nn.Module):
def __init__(self):
super().__init__()
# Downsampling
self.down1 = UNetBlock(3, 64)
self.down2 = UNetBlock(64, 128)
self.down3 = UNetBlock(128, 256)
# Bottleneck
self.bottleneck = UNetBlock(256, 512)
# Upsampling
self.up3 = UNetBlock(512, 256)
self.up2 = UNetBlock(256, 128)
self.up1 = UNetBlock(128, 64)
# Output
self.out = nn.Conv2d(64, 3, 1)
def forward(self, x, t):
# Embed timestep
t_emb = self.time_embedding(t)
# Downsample
d1 = self.down1(x, t_emb)
d2 = self.down2(d1, t_emb)
d3 = self.down3(d2, t_emb)
# Bottleneck
bottleneck = self.bottleneck(d3, t_emb)
# Upsample
u3 = self.up3(torch.cat([bottleneck, d3], dim=1), t_emb)
u2 = self.up2(torch.cat([u3, d2], dim=1), t_emb)
u1 = self.up1(torch.cat([u2, d1], dim=1), t_emb)
# Output
return self.out(u1)
The important thing components are:
- U-Net style architecture with skip connections
- Time embedding to condition on the timestep
- Flexible depth and width
Sampling Algorithm
Once we have trained our noise prediction network ε_θ, we are able to use it to generate recent samples. The essential sampling algorithm is:
- Start with pure Gaussian noise xT
- For t = T to 1:
- Predict noise:
ε_θ(x_t, t) - Compute mean:
μ = 1/√(1-β_t) * (x_t - β_t/√(1-α_t) * ε_θ(x_t, t)) - Sample:
x_{t-1} ~ N(μ, σ_t^2 * I)
- Predict noise:
- Return x₀
This process regularly denoises the sample, guided by our learned noise prediction network.
In practice, there are numerous sampling techniques that may improve quality or speed:
- DDIM sampling: A deterministic variant that enables for fewer sampling steps
- Ancestral sampling: Incorporates the learned variance σ_θ^2
- Truncated sampling: Stops early for faster generation
Here’s a basic implementation of the sampling algorithm:
def sample(model, n_samples, device):
# Start with pure noise
x = torch.randn(n_samples, 3, 32, 32).to(device)
for t in reversed(range(1000)):
# Add noise to create x_t
t_batch = torch.full((n_samples,), t, device=device)
noise = torch.randn_like(x)
x_t = add_noise(x, noise, t)
# Predict and take away noise
pred_noise = model(x_t, t_batch)
x = remove_noise(x_t, pred_noise, t)
# Add noise for next step (except at t=0)
if t > 0:
noise = torch.randn_like(x)
x = add_noise(x, noise, t-1)
return x
The Mathematics Behind Diffusion Models
To actually understand diffusion models, it’s crucial to delve deeper into the mathematics that underpin them. Let’s explore some key concepts in additional detail:
Markov Chain and Stochastic Differential Equations
The forward diffusion process in diffusion models could be viewed as a Markov chain or, in the continual limit, as a stochastic differential equation (SDE). The SDE formulation provides a strong theoretical framework for analyzing and increasing diffusion models.
The forward SDE could be written as:
dx = f(x,t)dt + g(t)dw
Where:
- f(x,t) is the drift term
- g(t) is the diffusion coefficient
- dw is a Wiener process (Brownian motion)
Different selections of f and g result in various kinds of diffusion processes. For instance:
- Variance Exploding (VE)
SDE: dx = √(d/dt σ²(t)) dw - Variance Preserving (VP)
SDE: dx = -0.5 β(t)xdt + √(β(t)) dw
Understanding these SDEs allows us to derive optimal sampling strategies and extend diffusion models to recent domains.
Rating Matching and Denoising Rating Matching
The connection between diffusion models and rating matching provides one other priceless perspective. The rating function is defined because the gradient of the log-probability density:
s(x) = ∇x log p(x)
Denoising rating matching goals to estimate this rating function by training a model to denoise barely perturbed data points. This objective seems to be akin to the diffusion model training objective in the continual limit.
This connection allows us to leverage techniques from score-based generative modeling, corresponding to annealed Langevin dynamics for sampling.
Advanced Training Techniques
Importance Sampling
The usual diffusion model training samples timesteps uniformly. Nevertheless, not all timesteps are equally necessary for learning. Importance sampling techniques could be used to focus training on essentially the most informative timesteps.
One approach is to make use of a non-uniform distribution over timesteps, weighted by the expected L2 norm of the rating:
p(t) ∝ E[||s(x_t, t)||²]
This could result in faster training and improved sample quality.
Progressive Distillation
Progressive distillation is a way to create faster sampling models without sacrificing quality. The method works as follows:
- Train a base diffusion model with many timesteps (e.g. 1000)
- Create a student model with fewer timesteps (e.g. 100)
- Train the coed to match the bottom model’s denoising process
- Repeat steps 2-3, progressively reducing timesteps
This permits for high-quality generation with significantly fewer denoising steps.
Architectural Innovations
Transformer-based Diffusion Models
While U-Net architectures have been popular for image diffusion models, recent work has explored using transformer architectures. Transformers offer several potential benefits:
- Higher handling of long-range dependencies
- More flexible conditioning mechanisms
- Easier scaling to larger model sizes
Models like DiT (Diffusion Transformers) have shown promising results, potentially offering a path to even higher quality generation.
Hierarchical Diffusion Models
Hierarchical diffusion models generate data at multiple scales, allowing for each global coherence and fine-grained details. The method typically involves:
- Generating a low-resolution output
- Progressively upsampling and refining
This approach could be particularly effective for high-resolution image generation or long-form content generation.
Advanced Topics
Classifier-Free Guidance
Classifier-free guidance is a way to enhance sample quality and controllability. The important thing idea is to coach two diffusion models:
- An unconditional model p(x_t)
- A conditional model p(x_t | y) where y is a few conditioning information (e.g. text prompt)
During sampling, we interpolate between these models:
ε_θ = (1 + w) * ε_θ(x_t | y) - w * ε_θ(x_t)
Where w > 0 is a guidance scale that controls how much to emphasise the conditional model.
This permits for stronger conditioning without having to retrain the model. It has been crucial for the success of text-to-image models like DALL-E 2 and Stable Diffusion.
Latent Diffusion
Source: Rombach et al.
Latent Diffusion Model (LDM) process involves encoding input data right into a latent space where the diffusion process occurs. The model progressively adds noise to the latent representation of the image, resulting in the generation of a loud version, which is then denoised using a U-Net architecture. The U-Net, guided by cross-attention mechanisms, integrates information from various conditioning sources like semantic maps, text, and image representations, ultimately reconstructing the image in pixel space. This process is pivotal in generating high-quality images with a controlled structure and desired attributes.
This offers several benefits:
- Faster training and sampling
- Higher handling of high-resolution images
- Easier to include conditioning
The method works as follows:
- Train an autoencoder to compress images to a latent space
- Train a diffusion model on this latent space
- For generation, sample in latent space and decode to pixels
This approach has been highly successful, powering models like Stable Diffusion.
Consistency Models
Consistency models are a recent innovation that goals to enhance the speed and quality of diffusion models. The important thing idea is to coach a single model that may map from any noise level on to the ultimate output, quite than requiring iterative denoising.
That is achieved through a rigorously designed loss function that enforces consistency between predictions at different noise levels. The result’s a model that may generate high-quality samples in a single forward pass, dramatically speeding up inference.
Practical Suggestions for Training Diffusion Models
Training high-quality diffusion models could be difficult. Listed here are some practical suggestions to enhance training stability and results:
- Gradient clipping: Use gradient clipping to forestall exploding gradients, especially early in training.
- EMA of model weights: Keep an exponential moving average (EMA) of model weights for sampling, which may result in more stable and higher-quality generation.
- Data augmentation: For image models, easy augmentations like random horizontal flips can improve generalization.
- Noise scheduling: Experiment with different noise schedules (linear, cosine, sigmoid) to seek out what works best in your data.
- Mixed precision training: Use mixed precision training to cut back memory usage and speed up training, especially for big models.
- Conditional generation: Even in case your end goal is unconditional generation, training with conditioning (e.g. on image classes) can improve overall sample quality.
Evaluating Diffusion Models
Properly evaluating generative models is crucial but difficult. Listed here are some common metrics and approaches:
Fréchet Inception Distance (FID)
FID is a widely used metric for evaluating the standard and variety of generated images. It compares the statistics of generated samples to real data within the feature space of a pre-trained classifier (typically InceptionV3).
Lower FID scores indicate higher quality and more realistic distributions. Nevertheless, FID has limitations and should not be the one metric used.
Inception Rating (IS)
Inception Rating measures each the standard and variety of generated images. It uses a pre-trained Inception network to compute:
IS = exp(E[KL(p(y|x) || p(y))])
Where p(y|x) is the conditional class distribution for generated image x.
Higher IS indicates higher quality and variety, but it surely has known limitations, especially for datasets very different from ImageNet.
Negative Log-likelihood (NLL)
For diffusion models, we are able to compute the negative log-likelihood of held-out data. This provides a direct measure of how well the model suits the true data distribution.
Nevertheless, NLL could be computationally expensive to estimate accurately for high-dimensional data.
Human Evaluation
For a lot of applications, especially creative ones, human evaluation stays crucial. This could involve:
- Side-by-side comparisons with other models
- Turing test-style evaluations
- Task-specific evaluations (e.g. image captioning for text-to-image models)
While subjective, human evaluation can capture features of quality that automated metrics miss.
Diffusion Models in Production
Deploying diffusion models in production environments presents unique challenges. Listed here are some considerations and best practices:
Optimization for Inference
- ONNX export: Convert models to ONNX format for faster inference across different hardware.
- Quantization: Use techniques like INT8 quantization to cut back model size and improve inference speed.
- Caching: For conditional models, cache intermediate results for the unconditional model to hurry up classifier-free guidance.
- Batch processing: Leverage batching to make efficient use of GPU resources.
Scaling
- Distributed inference: For prime-throughput applications, implement distributed inference across multiple GPUs or machines.
- Adaptive sampling: Dynamically adjust the variety of sampling steps based on the specified quality-speed tradeoff.
- Progressive generation: For giant outputs (e.g. high-res images), generate progressively from low to high resolution to supply faster initial results.
Safety and Filtering
- Content filtering: Implement robust content filtering systems to forestall generation of harmful or inappropriate content.
- Watermarking: Consider incorporating invisible watermarks into generated content for traceability.
Applications
Diffusion models have found success in a big selection of generative tasks:
Image Generation
Image generation is where diffusion models first gained prominence. Some notable examples include:
- DALL-E 3: OpenAI’s text-to-image model, combining a CLIP text encoder with a diffusion image decoder
- Stable Diffusion: An open-source latent diffusion model for text-to-image generation
- Imagen: Google’s text-to-image diffusion model
These models can generate highly realistic and inventive images from text descriptions, outperforming previous GAN-based approaches.
Video Generation
Diffusion models have also been applied to video generation:
- Video Diffusion Models: Generating video by treating time as an extra dimension within the diffusion process
- Make-A-Video: Meta’s text-to-video diffusion model
- Imagen Video: Google’s text-to-video diffusion model
These models can generate short video clips from text descriptions, opening up recent possibilities for content creation.
3D Generation
Recent work has prolonged diffusion models to 3D generation:
- DreamFusion: Text-to-3D generation using 2D diffusion models
- Point-E: OpenAI’s point cloud diffusion model for 3D object generation
These approaches enable the creation of 3D assets from text descriptions, with applications in gaming, VR/AR, and product design.
Challenges and Future Directions
While diffusion models have shown remarkable success, there are still several challenges and areas for future research:
Computational Efficiency
The iterative sampling strategy of diffusion models could be slow, especially for high-resolution outputs. Approaches like latent diffusion and consistency models aim to handle this, but further improvements in efficiency are an energetic area of research.
Controllability
While techniques like classifier-free guidance have improved controllability, there’s still work to be done in allowing more fine-grained control over generated outputs. This is very necessary for creative applications.
Multi-Modal Generation
Current diffusion models excel at single-modality generation (e.g. images or audio). Developing truly multi-modal diffusion models that may seamlessly generate across modalities is an exciting direction for future work.
Theoretical Understanding
While diffusion models have strong empirical results, there’s still more to grasp about why they work so well. Developing a deeper theoretical understanding may lead to further improvements and recent applications.
Conclusion
Diffusion models represent a step forward in generative AI, offering high-quality results across a variety of modalities. By learning to reverse a noise-adding process, they supply a versatile and theoretically grounded approach to generation.
From creative tools to scientific simulations, the flexibility to generate complex, high-dimensional data has the potential to remodel many fields. Nevertheless, it is vital to approach these powerful technologies thoughtfully, considering each their immense potential and the moral challenges they present.
