Random Walks with Tweedie: A Unified Framework for Diffusion Models

We present a simple template for designing generative diffusion model algorithms based on an interpretation of diffusion sampling as a sequence of random walks. Score-based diffusion models are widely used to generate high-quality images. Diffusion models have also been shown to yield state-of-the-art performance in many inverse problems. While these algorithms are often surprisingly simple, the theory behind them is not, and multiple complex theoretical justifications exist in the literature. Here, we provide a simple and largely self-contained theoretical justification for score-based-diffusion models that avoids using the theory of Markov chains or reverse diffusion, instead centering the theory of random walks and Tweedie's formula. This approach leads to unified algorithmic templates for network training and sampling. In particular, these templates cleanly separate training from sampling, e.g., the noise schedule used during training need not match the one used during sampling. We show that several existing diffusion models correspond to particular choices within this template and demonstrate that other, more straightforward algorithmic choices lead to effective diffusion models. The proposed framework has the added benefit of enabling conditional sampling without any likelihood approximation.