Submodularity of the expected information gain in infinite-dimensional linear inverse problems

TL;DR

This paper proves that the expected information gain (EIG) in infinite-dimensional linear Gaussian Bayesian inverse problems is submodular, enabling efficient greedy sensor placement strategies with theoretical guarantees.

Contribution

It extends the known submodularity of EIG from finite to infinite-dimensional inverse problems, ensuring optimal sensor placement methods remain effective.

Findings

EIG is submodular in infinite-dimensional settings.

Greedy sensor placement guarantees extend to infinite dimensions.

Strategies exploiting problem structure improve computational efficiency.

Abstract

We consider infinite-dimensional linear Gaussian Bayesian inverse problems with uncorrelated sensor data, and focus on the problem of finding sensor placements that maximize the expected information gain (EIG). This study is motivated by optimal sensor placement for linear inverse problems constrained by partial differential equations (PDEs). We consider measurement models where each sensor collects a single-snapshot measurement. This covers sensor placement for inverse problems governed by linear steady PDEs or evolution equations with final-in-time observations. It is well-known that in the finite-dimensional (discretized) formulations of such inverse problems, EIG is a monotone submodular function. This also entails a theoretical guarantee for greedy sensor placement in the discretized setting. We extend the result on submodularity of the EIG to the infinite-dimensional setting, proving that the approximation guarantee of greedy sensor placement remains valid in the infinite-dimensional limit. We also discuss computational considerations and present strategies that exploit problem structure and submodularity to yield an efficient implementation of the greedy procedure.