On submodularity of the expected information gain
Steven Maio, Alen Alexanderian
TL;DR
This paper addresses optimal experimental design for PDE-based linear Gaussian inverse problems with uncorrelated sensors. It shows the expected information gain, Phi_eig(S) = log det(I + tilde H(S)), is monotone and submodular, providing a simple PDE-weighted inner-product proof. The result relies on expressing the prior-preconditioned operator tilde H as a sum of rank-one terms and applying rank-one update identities. These insights justify greedy sensor placement with the classic (1 - 1/e) approximation guarantee and clarify the structure of information gains in discretized PDE settings.
Abstract
We consider finite-dimensional linear Gaussian Bayesian inverse problems with uncorrelated sensor measurements. In this setting, it is known that the expected information gain, quantified by the expected Kullback-Leibler divergence from the posterior measure to the prior measure, is submodular. We present a simple alternative proof of this fact tailored to a weighted inner product space setting arising from discretization of infinite-dimensional inverse problems constrained by partial differential equations (PDEs).