A Data-Adaptive Prior for Bayesian Learning of Kernels in Operators

TL;DR

This paper introduces a data-adaptive prior for Bayesian kernel learning in operators, ensuring stable posteriors under various errors and small noise, improving over fixed priors.

Contribution

The paper proposes a novel data-adaptive prior with an eigenspace-based covariance, enhancing stability and robustness in Bayesian kernel learning for operators.

Findings

Data-adaptive prior yields stable posterior means with small noise limits.

Fixed priors can diverge under discretization, model, partial observation, or wrong noise assumptions.

Numerical tests demonstrate the superiority of the data-adaptive prior in various error scenarios.

Abstract

Kernels are efficient in representing nonlocal dependence and they are widely used to design operators between function spaces. Thus, learning kernels in operators from data is an inverse problem of general interest. Due to the nonlocal dependence, the inverse problem can be severely ill-posed with a data-dependent singular inversion operator. The Bayesian approach overcomes the ill-posedness through a non-degenerate prior. However, a fixed non-degenerate prior leads to a divergent posterior mean when the observation noise becomes small, if the data induces a perturbation in the eigenspace of zero eigenvalues of the inversion operator. We introduce a data-adaptive prior to achieve a stable posterior whose mean always has a small noise limit. The data-adaptive prior's covariance is the inversion operator with a hyper-parameter selected adaptive to data by the L-curve method. Furthermore, we provide a detailed analysis on the computational practice of the data-adaptive prior, and demonstrate it on Toeplitz matrices and integral operators. Numerical tests show that a fixed prior can lead to a divergent posterior mean in the presence of any of the four types of errors: discretization error, model error, partial observation and wrong noise assumption. In contrast, the data-adaptive prior always attains posterior means with small noise limits.