Error Analysis of Bayesian Inverse Problems with Generative Priors

TL;DR

This paper provides a quantitative analysis of errors in Bayesian inverse problems using generative priors, establishing bounds and verifying them through numerical experiments including PDE inverse problems.

Contribution

It introduces error bounds for Wasserstein-2 generative models in Bayesian inverse problems and links prior errors to posterior errors under certain conditions.

Findings

Error bounds for Wasserstein-2 generative priors derived

Posterior error inherits prior error rate under assumptions

Numerical experiments confirm theoretical error bounds

Abstract

Data-driven methods for the solution of inverse problems have become widely popular in recent years thanks to the rise of machine learning techniques. A popular approach concerns the training of a generative model on additional data to learn a bespoke prior for the problem at hand. In this article we present an analysis for such problems by presenting quantitative error bounds for minimum Wasserstein-2 generative models for the prior. We show that under some assumptions, the error in the posterior due to the generative prior will inherit the same rate as the prior with respect to the Wasserstein-1 distance. We further present numerical experiments that verify that aspects of our error analysis manifests in some benchmarks followed by an elliptic PDE inverse problem where a generative prior is used to model a non-stationary field.