Optimal low-rank approximations of Bayesian linear inverse problems

TL;DR

This paper develops optimal low-rank approximation methods for Bayesian inverse problems, enabling efficient computation of posterior distributions by exploiting low-dimensional structures, with proven optimality and demonstrated effectiveness in high-dimensional applications.

Contribution

It introduces theoretically optimal low-rank updates for the posterior covariance and mean in Bayesian inverse problems, with fast approximations suitable for repeated evaluations.

Findings

Optimal low-rank covariance updates improve computational efficiency.

Fast, optimal posterior mean approximations for multiple data sets.

Numerical examples demonstrate near-full-space accuracy in high-dimensional problems.

Abstract

In the Bayesian approach to inverse problems, data are often informative, relative to the prior, only on a low-dimensional subspace of the parameter space. Significant computational savings can be achieved by using this subspace to characterize and approximate the posterior distribution of the parameters. We first investigate approximation of the posterior covariance matrix as a low-rank update of the prior covariance matrix. We prove optimality of a particular update, based on the leading eigendirections of the matrix pencil defined by the Hessian of the negative log-likelihood and the prior precision, for a broad class of loss functions. This class includes the F\"{o}rstner metric for symmetric positive definite matrices, as well as the Kullback-Leibler divergence and the Hellinger distance between the associated distributions. We also propose two fast approximations of the posterior mean and prove their optimality with respect to a weighted Bayes risk under squared-error loss. These approximations are deployed in an offline-online manner, where a more costly but data-independent offline calculation is followed by fast online evaluations. As a result, these approximations are particularly useful when repeated posterior mean evaluations are required for multiple data sets. We demonstrate our theoretical results with several numerical examples, including high-dimensional X-ray tomography and an inverse heat conduction problem. In both of these examples, the intrinsic low-dimensional structure of the inference problem can be exploited while producing results that are essentially indistinguishable from solutions computed in the full space.