On Provable Posterior Sampling with Denoising Oracles

Data Science Seminar

Joan Bruna
NYU Courant Institute

Abstract

Score-based diffusion models learn a denoising oracle (or score) from datasets that provides an efficient sampling scheme going beyond typical isoperimetric assumptions. They offer unprecedented ability to model complex data priors, used for solving inverse problems through posterior sampling. Although many heuristic methods have been developed recently for this purpose, they lack the quantitative guarantees needed in many scientific applications.

In this talk, focusing on linear inverse problems, we introduce the tilted transport technique. It leverages the quadratic structure of the log-likelihood in combination with the prior denoising oracle to transform the original posterior sampling problem into a new `boosted' posterior that is provably easier to sample from. Our analysis highlights the dependencies on the condition number of the measurement operator and the signal-to-noise ratio. The resulting posterior sampling scheme is shown to reach the computational threshold for sampling Ising models with a direct analysis, and is further validated on high-dimensional Gaussian mixture models and scalar field phi-4 models.

Joint work with Jiequn Han (Flatiron Institute).