Bayesian neural network priors for edge-preserving inversion
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.
Cite
@article{arxiv.2112.10663,
title = {Bayesian neural network priors for edge-preserving inversion},
author = {Chen Li and Matthew Dunlop and Georg Stadler},
journal= {arXiv preprint arXiv:2112.10663},
year = {2021}
}