Efficient sampling for sparse Bayesian learning using hierarchical prior normalization
For practitioners using Bayesian inference in high-dimensional inverse problems, this method improves MCMC efficiency by transforming sparsity-promoting priors into standard normal ones.
This paper introduces an efficient MCMC sampling method for sparse Bayesian learning by using hierarchical prior-normalizing transport maps, achieving significant performance improvements in inverse problems like signal deblurring and Burgers equation inversion.
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.