Local Diffusion Models and Phases of Data Distributions

As a class of generative artificial intelligence frameworks inspired by statistical physics, diffusion models have shown extraordinary performance in synthesizing complicated data distributions through a denoising process gradually guided by score functions. Real-life data, like images, is often spatially structured in low-dimensional spaces. However, ordinary diffusion models ignore this local structure and learn spatially global score functions, which are often computationally expensive. In this work, motivated by recent advances in non-equilibrium statistical physics, we develop a generic framework for defining phases of data distributions and use it to analyze the locality requirements of denoisers in diffusion models. We define two distributions as belonging to the same data distribution phase if they can be mutually connected via spatially local operations such as local denoisers, along the same evolution path as the diffusion. We demonstrate that the reverse denoising process consists of an early trivial phase and a late data phase, sandwiching a rapid phase transition where local denoisers must fail. We further demonstrate that the performance of local denoisers is closely tied to spatial Markovianity, which provides an operational criterion for diagnosing such phase transitions. We validate this criterion through numerical experiments on real-world datasets. Our work suggests guidance for simpler and more efficient architectures of diffusion models: far from the phase transition point, we can use small local neural networks to compute the score function; global neural networks are only necessary around the narrow time interval of phase transitions. This result also opens up new directions for studying phases of data distributions, the broader science of generative artificial intelligence, and guiding the design of neural networks inspired by physics concepts.