Efficient sampling for sparse Bayesian learning using hierarchical prior normalization

TL;DR

This paper presents a novel hierarchical prior-normalizing transport map approach that simplifies high-dimensional sparse Bayesian learning posteriors, enabling more efficient MCMC sampling and improved performance in inverse problems.

Contribution

Introduction of hierarchical prior-normalizing transport maps for efficient MCMC sampling in high-dimensional sparse Bayesian learning.

Findings

Significant performance improvements in MCMC sampling for inverse problems.

Effective transformation of complex priors into standard normal distributions.

Successful application to signal deblurring, Burgers equation inversion, and impulse image recovery.

Abstract

We introduce an approach for efficient Markov chain Monte Carlo (MCMC) sampling for challenging high-dimensional distributions in sparse Bayesian learning (SBL). The core innovation involves using hierarchical prior-normalizing transport maps (TMs), which are deterministic couplings that transform the sparsity-promoting SBL prior into a standard normal one. We analytically derive these prior-normalizing TMs by leveraging the product-like form of SBL priors and Knothe--Rosenblatt (KR) rearrangements. These transform the complex target posterior into a simpler reference distribution equipped with a standard normal prior that can be sampled more efficiently. Specifically, one can leverage the standard normal prior by using more efficient, structure-exploiting samplers. Our numerical experiments on various inverse problems -- including signal deblurring, inverting the non-linear inviscid Burgers equation, and recovering an impulse image -- demonstrate significant performance improvements for standard MCMC techniques.