On log-concave approximations of high-dimensional posterior measures and stability properties in non-linear inverse problems

TL;DR

This paper develops methods to approximate complex high-dimensional posterior measures with log-concave measures, enabling efficient sampling in nonlinear inverse problems with proven stability and convergence guarantees.

Contribution

It extends stability analysis of posterior measures and introduces a framework for approximating non-log-concave measures with log-concave ones in high dimensions.

Findings

Posterior measures can be approximated by log-concave measures in Wasserstein distance.

Gradient-based sampling algorithms are proven to converge polynomially under certain stability conditions.

New stability results are derived for a nonlinear inverse problem in integral geometry.

Abstract

The problem of efficiently generating random samples from high-dimensional and non-log-concave posterior measures arising from nonlinear regression problems is considered. Extending investigations from arXiv:2009.05298, local and global stability properties of the model are identified under which such posterior distributions can be approximated in Wasserstein distance by suitable log-concave measures. This allows the use of fast gradient based sampling algorithms, for which convergence guarantees are established that scale polynomially in all relevant quantities (assuming `warm' initialisation). The scope of the general theory is illustrated in a non-linear inverse problem from integral geometry for which new stability results are derived.