Understanding Diffusion Models: From Theory to Implementation

Introduction to Diffusion Models

Diffusion models have emerged as a powerful class of generative models in recent years. They are based on the idea of gradually adding noise to data and then learning to reverse this noise addition process. In this article, we'll dive deep into the theory behind diffusion models and explore how to implement them in practice.

Theoretical Foundations

The Forward Process

The forward process in a diffusion model is a fixed Markov chain that gradually adds Gaussian noise to the data. Starting from the original data x0, we apply T steps of noise addition to reach xT, which is pure noise.

The forward process is defined as:

xt = √(αt) * xt-1 + √(1 - αt) * ε

where ε ~ N(0, I) and αt are fixed parameters that control the amount of noise added at each step.

The Reverse Process

The goal is to learn a reverse process that can generate data by gradually denoising pure noise. This reverse process is modeled as:

p(xt-1 | xt) = N(xt-1; μθ(xt, t), σθ^2(xt, t)I)

where μθ and σθ are learned functions parameterized by a neural network.

ELBO Optimization

To train the model, we optimize the evidence lower bound (ELBO):

ELBO = E[log p(x0|x1) - KL(q(xT|x0) || p(xT)) - Σ KL(q(xt-1|xt,x0) || p(xt-1|xt))]

This can be simplified to:

ELBO ≈ E[log p(x0|x1) - ||εθ(xt,t) - ε||^2]

where εθ is a neural network that predicts the noise added at each step.

Practical Implementation

Neural Network Architecture

The core of a diffusion model is a neural network εθ(xt,t) that takes a noisy sample xt and the timestep t as input, and predicts the noise that was added. A common architecture is a U-Net with time embedding:

1. Input layer: Takes xt (e.g. a noisy image)
2. Time embedding: Encodes t using sinusoidal embeddings
3. U-Net backbone: Processes the input with skip connections
4. Output layer: Predicts the noise ε

Training Loop

The training process involves:

1. Sample a batch of data x0
2. Sample timesteps t uniformly
3. Sample noise ε
4. Compute noisy samples xt = √(αt_bar) * x0 + √(1 - αt_bar) * ε
5. Predict noise εθ(xt, t)
6. Compute loss L = ||εθ(xt, t) - ε||^2
7. Update network parameters

Sampling

To generate new samples:

1. Start with pure noise xT ~ N(0, I)
2. Iteratively denoise: xt-1 = 1/√αt * (xt - √(1-αt) * εθ(xt, t)) + σt * z where z ~ N(0, I) and σt is a small noise term
3. Repeat until x0 is obtained

Advanced Topics

Conditional Generation

Diffusion models can be extended to conditional generation by incorporating class information or text embeddings into the denoising network. This allows generating images based on text prompts or class labels.

Latent Diffusion Models

To improve efficiency, diffusion can be performed in a learned latent space rather than pixel space. This is the approach used in state-of-the-art text-to-image models like Stable Diffusion.

Consistency Models

Recent work has shown that diffusion models can be distilled into deterministic models that generate high-quality samples in a single forward pass, dramatically speeding up inference.

Conclusion

Diffusion models have revolutionized generative AI, enabling high-quality image synthesis, inpainting, super-resolution, and more. By understanding the mathematical foundations and implementation details, researchers and practitioners can leverage these powerful models for a wide range of applications. As the field continues to advance, we can expect even more impressive capabilities from diffusion-based generative AI in the coming years.

Article created from: https://youtu.be/D-JQVOodqmg?feature=shared