Following our study of Generative Adversarial Networks, this chapter introduces Diffusion Models, a distinct and effective approach to generative modeling. These models have demonstrated remarkable success in generating high-quality data, especially images, by progressively adding noise to data points and then learning the reverse denoising process.

This chapter covers the following key areas:

• The mathematical basis connecting diffusion processes to Stochastic Differential Equations ($SDE$s).
• The formulation and implementation details of Denoising Diffusion Probabilistic Models (DDPMs), including their specific objective functions, such as the simplified objective often used in practice: $L_{simple}(\theta) = \mathbb{E}_{t, \mathbf{x}_0, \boldsymbol{\epsilon}} \left[ \| \boldsymbol{\epsilon} - \boldsymbol{\epsilon}_\theta(\sqrt{\bar{\alpha}_t}\mathbf{x}_0 + \sqrt{1-\bar{\alpha}_t}\boldsymbol{\epsilon}, t) \|^2 \right]$.
• The relationship between diffusion models and Score-Based Generative Modeling via concepts like score matching and Langevin dynamics.
• Methods for accelerating the sampling process, such as Denoising Diffusion Implicit Models (DDIM), and techniques for controlling the generation using classifier guidance or classifier-free guidance.
• Common neural network architectures employed, particularly the U-Net structure adapted for noise prediction.
• Building a foundational DDPM through a hands-on coding section to solidify understanding.

You will gain an understanding of the theory behind diffusion models and the practical considerations for implementing and improving them effectively.