Diffusion Models

Introduction to Diffusion Models

Diffusion models are a class of generative models inspired by non-equilibrium thermodynamics. They learn to generate data by reversing a gradual noising process. This approach has shown remarkable success in generating high-quality images, audio, and other complex data.

Forward Diffusion Process (Noising)

The forward diffusion process gradually adds Gaussian noise to the input data over a series of time steps $T$. Starting from a real data sample $x_0 \sim q(x)$, a Markov chain of diffusion steps is defined, producing a sequence of noisy samples $x_1, x_2, ..., x_T$. Each step adds a small amount of Gaussian noise according to a variance schedule $\{\beta_t\}_{t=1}^T$. The conditional distribution for the forward process is:

\(q(x_t | x_{t-1}) = \mathcal{N}(x_t; \sqrt{1 - \beta_t} x_{t-1}, \beta_t \mathbf{I})\)

For sufficiently large $T$ and a well-designed variance schedule, the data distribution $q(x_T)$ approaches a simple, tractable distribution, typically an isotropic Gaussian distribution $\mathcal{N}(0, \mathbf{I})$.

Reverse Diffusion Process (Denoising)

The goal is to learn the reverse process, which starts from the noise $x_T \sim \mathcal{N}(0, \mathbf{I})$ and gradually denoises it back to a real data sample $x_0$. If we knew the conditional probability $p(x_{t-1} | x_t)$, we could sample from the reverse process. This reverse process is also a Gaussian distribution:

\(p(x_{t-1} | x_t) = \mathcal{N}(x_{t-1}; \mu_\theta(x_t, t), \Sigma_\theta(x_t, t))\)

A neural network is trained to predict the parameters of this conditional distribution, $\mu_\theta(x_t, t)$ (mean) and $\Sigma_\theta(x_t, t)$ (covariance), at each time step $t$. The training objective typically involves predicting the noise added at each step of the forward process.

Key Concepts and Architectures

DDPM (Denoising Diffusion Probabilistic Models)

DDPM, as introduced by Ho et al. (2020), is a foundational diffusion model that defines the forward and reverse processes as described above. It trains a U-Net architecture to predict the noise added at each diffusion step. The loss function aims to minimize the difference between the predicted noise and the actual noise added during the forward process.

Stable Diffusion

Stable Diffusion, developed by Rombach et al. (2022), is a significant advancement that operates in the latent space of a pre-trained autoencoder. This reduces the dimensionality of the data, making the diffusion process more computationally efficient and allowing for the generation of high-resolution images with lower computational resources. Text-to-image generation is achieved by conditioning the denoising process on text embeddings.

Noise Scheduling (Variance Schedule)

The variance schedule $\{\beta_t\}_{t=1}^T$ in the forward diffusion process determines how much noise is added at each time step. Different schedules (e.g., linear, cosine) can significantly impact the training and generation quality. Carefully designed schedules help in ensuring a smooth transition to a Gaussian distribution and facilitate effective learning of the reverse process.

Latent Diffusion Models (LDMs)

Latent Diffusion Models (LDMs), as exemplified by Stable Diffusion, perform the diffusion and denoising processes in a lower-dimensional latent space learned by an autoencoder. This approach offers several advantages, including reduced computational cost, faster sampling, and the ability to condition generation on various inputs (e.g., text, semantic maps) more effectively.

Score-Based Generative Models

Score-based generative models are closely related to diffusion models. They aim to learn the score function (the gradient of the log probability density) of the data distribution at different noise levels. Sampling is then performed by following the gradient of the score function back to the data manifold using Langevin dynamics or similar techniques. DDPM can be viewed as a specific parameterization of a score-based model.

Denoising Steps

The reverse diffusion process involves iteratively denoising a noisy sample over multiple steps. At each step $t$, the trained neural network predicts the mean (and sometimes the variance) of the previous step's distribution, allowing us to sample a less noisy version of the data. The quality of the generated data heavily depends on the number of denoising steps and the accuracy of the learned denoising function.

Applications and Advancements

Diffusion models have revolutionized generative modeling and are being applied to a wide range of tasks:

 
• Image Synthesis: Generating photorealistic images from text prompts (e.g., Stable Diffusion, DALL-E 2, Midjourney).
 
• Image Editing: Performing semantically meaningful edits on existing images.
 
• Video Generation: Creating realistic and coherent video sequences.
 
• Audio Synthesis: Generating high-fidelity audio, music, and speech.
 
• 3D Generation: Creating 3D models and scenes.
 
• Scientific Applications: Generating molecules, materials, and biological sequences.

Ongoing research focuses on improving the efficiency (reducing the number of denoising steps), controllability (better alignment with conditioning inputs), and fidelity of diffusion models.

Challenges and Future Directions

Despite their success, diffusion models still face challenges:

 
• Computational Cost: Training and sampling can be computationally expensive, especially for high-resolution data and a large number of denoising steps.
 
• Sampling Speed: Generating a single high-quality sample can take significant time due to the iterative denoising process.
 
• Controllability: Achieving fine-grained control over the generation process can be challenging.

Future research directions include:

 
• Faster Sampling Techniques: Developing methods to reduce the number of denoising steps without sacrificing quality (e.g., using advanced ODE solvers or distillation techniques).
 
• Improved Efficiency: Designing more efficient model architectures and training strategies.
 
• Enhanced Controllability: Exploring better ways to condition the generation process on various modalities and fine-grained instructions.
 
• Theoretical Understanding: Further investigating the theoretical foundations of diffusion models to guide architectural design and training.