Core Viewpoint - The article discusses the mathematical principles behind diffusion models, emphasizing the importance of noise in the sampling process and how it contributes to generating diverse and realistic images. The key takeaway is that diffusion models leverage Langevin sampling to transition from one probability distribution to another, with noise being an essential component rather than a mere side effect [10][11][26]. Summary by Sections Section 1: Basic Concepts of Diffusion Models - The article introduces the foundational concepts related to diffusion models, focusing on the use of velocity vector fields to define ordinary differential equations (ODEs) and the mathematical representation of these fields through trajectories [4]. Section 2: Langevin Sampling - Langevin sampling is highlighted as a crucial method for approximating transitions between distributions. The process involves adding noise to the sampling, which allows for exploration of the probability space and prevents convergence to local maxima [10][11][14][26]. Section 3: Role of Noise - Noise is described as a necessary component in the diffusion process, enabling the model to generate diverse samples rather than converging to peak values. The article explains that without noise, the sampling process would only yield local maxima, limiting the diversity of generated outputs [26][28][31]. Section 4: Comparison with GANs - The article contrasts diffusion models with Generative Adversarial Networks (GANs), noting that diffusion models assign the task of diversity to noise, which alleviates issues like mode collapse that can occur in GANs [37]. Section 5: Training and Implementation - The training process for diffusion models involves using score matching and kernel density estimation (KDE) to learn the underlying data distribution. The article outlines the steps for training, including the generation of noisy samples and the calculation of gradients for optimization [64][65]. Section 6: Flow Matching Techniques - Flow matching is introduced as a method for optimizing the sampling process, with a focus on minimizing the distance between the learned velocity field and the true data distribution. The article discusses the equivalence of flow matching and optimal transport strategies [76][86]. Section 7: Mean Flow and Rectified Flow - Mean flow and rectified flow are presented as advanced techniques within the flow matching framework, emphasizing their ability to improve sampling efficiency and stability during the generation process [100][106].