Posterior-Variance-Based Error Quantification for Inverse Problems in Imaging

TL;DR

This paper introduces a Bayesian error quantification method for inverse imaging problems that provides pixel-wise error bounds with coverage guarantees, applicable to various priors and approximate sampling, validated through experiments and a new sampling algorithm.

Contribution

It presents a novel approach combining posterior variance estimates with conformal prediction to achieve distribution-free coverage guarantees in Bayesian inverse problems.

Findings

Error bounds are tight in practice.

Method is applicable with approximate posterior sampling.

Introduces a new primal-dual Langevin sampling algorithm.

Abstract

In this work, a method for obtaining pixel-wise error bounds in Bayesian regularization of inverse imaging problems is introduced. The proposed method employs estimates of the posterior variance together with techniques from conformal prediction in order to obtain coverage guarantees for the error bounds, without making any assumption on the underlying data distribution. It is generally applicable to Bayesian regularization approaches, independent, e.g., of the concrete choice of the prior. Furthermore, the coverage guarantees can also be obtained in case only approximate sampling from the posterior is possible. With this in particular, the proposed framework is able to incorporate any learned prior in a black-box manner. Guaranteed coverage without assumptions on the underlying distributions is only achievable since the magnitude of the error bounds is, in general, unknown in advance. Nevertheless, experiments with multiple regularization approaches presented in the paper confirm that in practice, the obtained error bounds are rather tight. For realizing the numerical experiments, also a novel primal-dual Langevin algorithm for sampling from non-smooth distributions is introduced in this work.