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

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.

💡 Research Summary

This paper addresses the optimal sensor placement problem for infinite‑dimensional linear Gaussian Bayesian inverse problems, focusing on maximizing the expected information gain (EIG). The authors consider a parameter m belonging to a real separable Hilbert space M, a linear observation operator F: M → ℝᵈ, and independent Gaussian noise η with covariance Γ_noise = diag(σ₁²,…,σ_d²). A Gaussian prior µ_pr = N(m_pr, C_pr) is assumed, where C_pr is a positive, self‑adjoint, trace‑class operator. Because the forward model is linear and the noise is Gaussian, the posterior is also Gaussian with covariance
C_post = (F* Γ_noise⁻¹ F + C_pr⁻¹)⁻¹.

The expected information gain, defined as the expected Kullback‑Leibler divergence from posterior to prior, admits a closed‑form expression that extends to infinite dimensions:
EIG(S) = ½ log det(I + Ĥ(S)),
where Ĥ(S) = C_pr^{½} H(S) C_pr^{½} and H(S) = ∑_{i∈S} f_i⊗f_i with f_i = σ_i⁻¹ F* e_i. The set S ⊆ {1,…,d} denotes the chosen sensor locations. Thus, for any design S, the posterior covariance can be written as
C_post(S) = C_pr^{½} (I + Ĥ(S))⁻¹ C_pr^{½}.

A key technical contribution is an infinite‑dimensional analogue of the Sherman‑Morrison‑Woodbury formula (Lemma 3.2). It shows that for a bijective bounded operator A and vectors u, v, the rank‑one update A + u⊗v is invertible with an explicit inverse expression. Applying this to A = I + Ĥ(S) yields a simple update for the posterior covariance when a new sensor i is added:
C_post(S∪{i}) = C_post(S) −