Posterior-Variance-Based Error Quantification for Inverse Problems in Imaging
This provides a method for reliable error quantification in imaging applications, though it is incremental as it builds on existing Bayesian and conformal prediction techniques.
The paper tackles the problem of quantifying pixel-wise error bounds in Bayesian regularization for inverse imaging problems, achieving coverage guarantees without distributional assumptions and demonstrating that the error bounds are tight in practice.
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.