# Sparsity enforcing priors in inverse problems via Normal variance mixtures: model selection, algorithms and applications

## TL;DR

This paper develops hierarchical Bayesian models using Normal variance mixtures to enforce sparsity in inverse problems, deriving algorithms for estimation and comparing their performance in applications like 3D-CT and chronobiology.

## Contribution

It introduces a unified framework for sparsity enforcing priors via Normal variance mixtures and develops iterative algorithms for inverse problems, including theoretical comparisons with regularization methods.

## Key findings

- Algorithms perform well in 3D-CT and chronobiology applications.
- Bayesian sparsity algorithms compare favorably with regularization techniques.
- Hierarchical models effectively incorporate uncertainties in inverse problems.

## Abstract

The sparse structure of the solution for an inverse problem can be modelled using different sparsity enforcing priors when the Bayesian approach is considered. Analytical expression for the unknowns of the model can be obtained by building hierarchical models based on sparsity enforcing distributions expressed via conjugate priors. We consider heavy tailed distributions with this property: the Student-t distribution, which is expressed as a Normal scale mixture, with the mixing distribution the Inverse Gamma distribution, the Laplace distribution, which can also be expressed as a Normal scale mixture, with the mixing distribution the Exponential distribution or can be expressed as a Normal inverse scale mixture, with the mixing distribution the Inverse Gamma distribution, the Hyperbolic distribution, the Variance-Gamma distribution, the Normal-Inverse Gaussian distribution, all three expressed via conjugate distributions using the Generalized Hyperbolic distribution. For all distributions iterative algorithms are derived based on hierarchical models that account for the uncertainties of the forward model. For estimation, Maximum A Posterior (MAP) and Posterior Mean (PM) via variational Bayesian approximation (VBA) are used. The performances of resulting algorithm are compared in applications in 3D computed tomography (3D-CT) and chronobiology. Finally, a theoretical study is developed for comparison between sparsity enforcing algorithms obtained via the Bayesian approach and the sparsity enforcing algorithms issued from regularization techniques, like LASSO and some others.