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

Andreas Habring, Thomas Pock, Martin Holler, Dominik Narnhofer

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.