Bayesian neural network priors for edge-preserving inversion

TL;DR

This paper introduces neural network-based priors with heavy-tailed weights for Bayesian inverse problems, effectively capturing discontinuities and edges in functions, and demonstrates improved accuracy and uncertainty quantification in deconvolution tasks.

Contribution

It proposes a novel class of neural network priors with heavy-tailed weights for edge-preserving Bayesian inversion, supported by theoretical and numerical evidence.

Findings

Heavy-tailed neural network priors produce discontinuous-like samples.

Point estimates with these priors outperform non-heavy tailed ones.

Uncertainty quantification is more informative with the proposed priors.

Abstract

We consider Bayesian inverse problems wherein the unknown state is assumed to be a function with discontinuous structure a priori. A class of prior distributions based on the output of neural networks with heavy-tailed weights is introduced, motivated by existing results concerning the infinite-width limit of such networks. We show theoretically that samples from such priors have desirable discontinuous-like properties even when the network width is finite, making them appropriate for edge-preserving inversion. Numerically we consider deconvolution problems defined on one- and two-dimensional spatial domains to illustrate the effectiveness of these priors; MAP estimation, dimension-robust MCMC sampling and ensemble-based approximations are utilized to probe the posterior distribution. The accuracy of point estimates is shown to exceed those obtained from non-heavy tailed priors, and uncertainty estimates are shown to provide more useful qualitative information.