Diffusion Models: DDPM数学推导与图像生成算法分析

enoising Diffusion Probabilistic Models) model proposed in 2020, this article provides a detailed introduction to the principles of diffusion models. For the diffusion process, under the condition of satisfying the Markov chain, we use reparameterization techniques to prove that we can sample at any step based on the original data, and can express it using mathematical expressions. For the inverse diffusion process, we apply Bayesian formula and the properties of Gaussian distribution to mathematically derive the posterior distribution of the inverse diffusion process, and apply it to target optimization. In terms of target optimization, under the condition of known real data, we use maximum likelihood estimation to transform the model parameter estimation into log-likelihood estimation, combined with variational inference and KL divergence to transform maximizing log-likelihood into minimizing the variational lower bound problem. Through mathematical derivation, the original predicted mean is transformed into predicted noise, simplifying the model's target optimization. After obtaining the objective function, we complete the algorithm flow introduction of model design and provide training models. Finally, based on the diffusion model, we implemented code for unconditional image generation and conditional image generation, and analyzed the results. Keywords: DDPM, diffusion model, reparameterization technique, variational inference, KL divergence.