This tutorial, authored by Stanley Chan, provides an in-depth exploration of diffusion models in imaging and vision applications. The primary goal is to explain the fundamental concepts behind diffusion models, making it accessible to undergraduate and graduate students interested in research or practical applications. 1. **Variational Auto-Encoder (VAE)** - **Setting**: Introduces the basic structure of a VAE, including the encoder and decoder, and the role of proxy distributions \( q_\phi(\mathbf{z}|\mathbf{x}) \) and \( p_\theta(\mathbf{x}|\mathbf{z}) \). - **Evidence Lower Bound (ELBO)**: Derives the ELBO as a lower bound for the log-likelihood of the data, explaining how it is used to train the VAE. - **Training VAE**: Details the training process, including the use of neural networks to estimate the mean and variance of the Gaussian distributions. - **Inference with VAE**: Explains how to use the trained VAE for inference by generating images from latent vectors. 2. **Denoising Diffusion Probabilistic Model (DDPM)** - **Building Blocks**: Describes the structure of the DDPM, including the transition blocks, initial block, and final block. - **Transition Distribution**: Introduces the transition distribution \( q_{\phi}(\mathbf{x}_t | \mathbf{x}_{t-1}) \) and explains why the scalars \( \sqrt{\alpha_t} \) and \( 1 - \alpha_t \) are chosen. - **Conditional Distribution**: Derives the conditional distribution \( q_{\phi}(\mathbf{x}_t | \mathbf{x}_0) \) and its utility in forward diffusion steps. - **Evidence Lower Bound**: Formulates the ELBO for the DDPM, explaining its components: reconstruction, prior matching, and consistency. 3. **Score-Matching Langevin Dynamics (SMLD)** - **Langevin Dynamics**: Introduces Langevin dynamics and its role in the diffusion process. - **Score Function**: Explains the concept of the score function and its importance in SMLD. - **Score Matching Techniques**: Discusses various techniques for score matching in SMLD. 4. **Stochastic Differential Equation (SDE)** - **Motivating Examples**: Provides examples to motivate the use of SDEs in diffusion models. - **Forward and Backward Iterations**: Explains the forward and backward iterations in SDEs. - **SDE for DDPM and SMLD**: Derives the SDEs for DDPM and SMLD. - **Solving SDE**: Discusses methods for solving SDEs. 5. **Conclusion** - Summarizes the key points covered in the tutorial and highlights the importance ofThis tutorial, authored by Stanley Chan, provides an in-depth exploration of diffusion models in imaging and vision applications. The primary goal is to explain the fundamental concepts behind diffusion models, making it accessible to undergraduate and graduate students interested in research or practical applications. 1. **Variational Auto-Encoder (VAE)** - **Setting**: Introduces the basic structure of a VAE, including the encoder and decoder, and the role of proxy distributions \( q_\phi(\mathbf{z}|\mathbf{x}) \) and \( p_\theta(\mathbf{x}|\mathbf{z}) \). - **Evidence Lower Bound (ELBO)**: Derives the ELBO as a lower bound for the log-likelihood of the data, explaining how it is used to train the VAE. - **Training VAE**: Details the training process, including the use of neural networks to estimate the mean and variance of the Gaussian distributions. - **Inference with VAE**: Explains how to use the trained VAE for inference by generating images from latent vectors. 2. **Denoising Diffusion Probabilistic Model (DDPM)** - **Building Blocks**: Describes the structure of the DDPM, including the transition blocks, initial block, and final block. - **Transition Distribution**: Introduces the transition distribution \( q_{\phi}(\mathbf{x}_t | \mathbf{x}_{t-1}) \) and explains why the scalars \( \sqrt{\alpha_t} \) and \( 1 - \alpha_t \) are chosen. - **Conditional Distribution**: Derives the conditional distribution \( q_{\phi}(\mathbf{x}_t | \mathbf{x}_0) \) and its utility in forward diffusion steps. - **Evidence Lower Bound**: Formulates the ELBO for the DDPM, explaining its components: reconstruction, prior matching, and consistency. 3. **Score-Matching Langevin Dynamics (SMLD)** - **Langevin Dynamics**: Introduces Langevin dynamics and its role in the diffusion process. - **Score Function**: Explains the concept of the score function and its importance in SMLD. - **Score Matching Techniques**: Discusses various techniques for score matching in SMLD. 4. **Stochastic Differential Equation (SDE)** - **Motivating Examples**: Provides examples to motivate the use of SDEs in diffusion models. - **Forward and Backward Iterations**: Explains the forward and backward iterations in SDEs. - **SDE for DDPM and SMLD**: Derives the SDEs for DDPM and SMLD. - **Solving SDE**: Discusses methods for solving SDEs. 5. **Conclusion** - Summarizes the key points covered in the tutorial and highlights the importance of