A Variational Inference Approach to Inverse Problems with Gamma Hyperpriors

TL;DR

This paper introduces a variational inference method for hierarchical Bayesian inverse problems with gamma hyperpriors, enabling accurate reconstructions and uncertainty quantification, surpassing traditional MAP estimation.

Contribution

It presents a novel variational iterative scheme for gamma hyperprior models, facilitating uncertainty quantification and hyperparameter model selection in inverse problems.

Findings

Accurate reconstructions demonstrated in deconvolution and dynamical systems examples.

Method provides meaningful uncertainty quantification.

Easy to implement and adaptable to hyperparameter tuning.

Abstract

Hierarchical models with gamma hyperpriors provide a flexible, sparse-promoting framework to bridge $L^1$ and $L^2$ regularizations in Bayesian formulations to inverse problems. Despite the Bayesian motivation for these models, existing methodologies are limited to \textit{maximum a posteriori} estimation. The potential to perform uncertainty quantification has not yet been realized. This paper introduces a variational iterative alternating scheme for hierarchical inverse problems with gamma hyperpriors. The proposed variational inference approach yields accurate reconstruction, provides meaningful uncertainty quantification, and is easy to implement. In addition, it lends itself naturally to conduct model selection for the choice of hyperparameters. We illustrate the performance of our methodology in several computed examples, including a deconvolution problem and sparse identification of dynamical systems from time series data.