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arXiv · 2112.10663

Bayesian neural network priors for edge-preserving inversion

Chen Li·Matthew Dunlop·Georg Stadler
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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.

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Chen Li, Matthew Dunlop, Georg Stadler. 2021-12-20. Bayesian neural network priors for edge-preserving inversion. https://arxiv.org/abs/2112.10663

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