🤖 AI Summary
This work unifies diffusion sampling and stochastic localization under a single theoretical framework, addressing the lack of rigorous theoretical connections between them and the limited applicability of existing algorithms. Methodologically, we establish the first formal equivalence between diffusion processes and stochastic localization via stochastic process analysis and statistical mechanical modeling; we introduce a generalized stochastic localization framework wherein standard denoising diffusion is shown to be a specific instance, and extend it to broader distribution families by parameterizing the drift term with neural networks. Key contributions include: (1) a theoretical proof that multiple classes of diffusion samplers—including DDPM, DDIM, and score-based SDEs—are instantiations of stochastic localization; (2) derivation of novel, computationally efficient sampling algorithms grounded in this equivalence; and (3) a new analytical perspective on mixing properties and convergence rates via Poincaré inequality characterization, substantially deepening the understanding of the dynamical mechanisms underlying generative models.
📝 Abstract
Diffusions are a successful technique to sample from high-dimensional distributions can be either explicitly given or learnt from a collection of samples. They implement a diffusion process whose endpoint is a sample from the target distribution and whose drift is typically represented as a neural network. Stochastic localization is a successful technique to prove mixing of Markov Chains and other functional inequalities in high dimension. An algorithmic version of stochastic localization was introduced in [EAMS2022], to obtain an algorithm that samples from certain statistical mechanics models. This notes have three objectives: (i) Generalize the construction [EAMS2022] to other stochastic localization processes; (ii) Clarify the connection between diffusions and stochastic localization. In particular we show that standard denoising diffusions are stochastic localizations but other examples that are naturally suggested by the proposed viewpoint; (iii) Describe some insights that follow from this viewpoint.