Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions

Score-based generative models (SGMs) are a family of generative models which achieve state-of-the-art performance for generating audio and image data [Soh+15; HJA20; DN21; Kin+21; Son+21a; Son+21b; VKK21]; see, e.g., the recent surveys [Cao+22; Cro+22; Yan+22]. One notable example of an SGM are denoising diffusion probabilistic models (DDPMs) [Soh+15; HJA20], which are a key component in largescale generative models such as DALL E 2 [Ram+22]. As the importance of SGMs continues to grow due to newfound applications in commercial domains, it is a pressing question of both practical and theoretical concern to understand the mathematical underpinnings which explain their startling empirical successes. As we explain in more detail in Section 2, at their mathematical core, SGMs consist of two stochastic processes, which we call the forward process and the reverse process. The forward process transforms samples from a data distribution q (e.g., natural images) into pure noise, whereas the reverse process transforms pure noise into samples from q, hence performing generative modeling. Implementation of the reverse process requires estimation of the score function of the law of the forward process, which is typically accomplished by training neural networks on a score matching objective [Hyv05; Vin11; SE19]. Providing precise guarantees for estimation of the score function is difficult, as it requires an understanding of the non-convex training dynamics of neural network optimization that is currently out of reach. However, given the empirical success of neural networks on the score estimation task, a natural and important question is whether or not accurate score estimation implies that SGMs provably converge to the true data distribution in realistic settings.