A variational inference framework for inverse problems

TL;DR

This paper introduces a flexible variational inference framework for inverse problems, enabling efficient and accurate model fitting across various applications, including image processing and biomedical simulations.

Contribution

It develops a variational Bayes approach with message passing and factor graphs, enhancing implementation efficiency and flexibility for inverse problems in multiple dimensions.

Findings

Efficient variational Bayes algorithms outperform MCMC in computational speed.

Framework supports diverse response distributions and penalizations.

Applicable to 1D, 2D, and higher-dimensional inverse problems.

Abstract

A framework is presented for fitting inverse problem models via variational Bayes approximations. This methodology guarantees flexibility to statistical model specification for a broad range of applications, good accuracy and reduced model fitting times. The message passing and factor graph fragment approach to variational Bayes that is also described facilitates streamlined implementation of approximate inference algorithms and allows for supple inclusion of numerous response distributions and penalizations into the inverse problem model. Models for one- and two-dimensional response variables are examined and an infrastructure is laid down where efficient algorithm updates based on nullifying weak interactions between variables can also be derived for inverse problems in higher dimensions. An image processing application and a simulation exercise motivated by biomedical problems reveal the computational advantage offered by efficient implementation of variational Bayes over Markov chain Monte Carlo.