Sampling, Diffusions, and Stochastic Localization
Andrea Montanari
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.