Fast sampling in a linear-Gaussian inverse problem

TL;DR

This paper demonstrates that efficient Bayesian sampling methods can outperform traditional regularized inversion in speed for deblurring inverse problems, enabling faster and scalable image reconstruction.

Contribution

It introduces a novel sampling approach that outperforms regularized inversion and common MCMC methods in speed and scalability for inverse problems.

Findings

Sampling the marginal posterior for hyperparameters and then the full conditional for the image is faster than regularized inversion.

The proposed method outperforms random-walk Metropolis-Hastings and block Gibbs MCMC in scaling with problem size.

Sample-based Bayesian inference can be performed with cost comparable to a single linear solve, enabling direct function space inference.

Abstract

We solve the inverse problem of deblurring a pixelized image of Jupiter using regularized deconvolution and by sample-based Bayesian inference. By efficiently sampling the marginal posterior distribution for hyperparameters, then the full conditional for the deblurred image, we find that we can evaluate the posterior mean faster than regularized inversion, when selection of the regularizing parameter is considered. To our knowledge, this is the first demonstration of sampling and inference that takes less compute time than regularized inversion in an inverse problems. Comparison to random-walk Metropolis-Hastings and block Gibbs MCMC shows that marginal then conditional sampling also outperforms these more common sampling algorithms, having better scaling with problem size. When problem-specific computations are feasible the asymptotic cost of an independent sample is one linear solve, implying that sample-based Bayesian inference may be performed directly over function spaces, when that limit exists.