Posterior contraction for empirical Bayesian approach to inverse problems under non-diagonal assumption

TL;DR

This paper develops an empirical Bayesian method for linear inverse problems with Gaussian priors, demonstrating automatic hyperparameter selection and near-optimal contraction rates without requiring common basis assumptions.

Contribution

It introduces a novel adaptive Bayesian approach that works under non-diagonal assumptions, broadening applicability in inverse problems.

Findings

Achieves near-optimal contraction rates without prior regularity knowledge.

Automatically selects hyperparameters in Gaussian inverse problems.

Applicable to a wider class of problems beyond common basis assumptions.

Abstract

We investigate an empirical Bayesian nonparametric approach to a family of linear inverse problems with Gaussian prior and Gaussian noise. We consider a class of Gaussian prior probability measures with covariance operator indexed by a hyperparameter that quantifies regularity. By introducing two auxiliary problems, we construct an empirical Bayes method and prove that this method can automatically select the hyperparameter. In addition, we show that this adaptive Bayes procedure provides optimal contraction rates up to a slowly varying term and an arbitrarily small constant, without knowledge about the regularity index. Our method needs not the prior covariance, noise covariance and forward operator have a common basis in their singular value decomposition, enlarging the application range compared with the existing results.