Measurement Clusterization in D-optimal Designs for Bayesian Linear Inverse Problems over Hilbert Spaces

Inversion for parameters of a physical process typically requires taking expensive measurements, and the task of finding an optimal set of measurements is known as the optimal design problem. Surprisingly, measurement locations in optimal designs are sometimes extremely clustered, and researchers often avoid measurement clusterization by modifying the optimal design problem. We consider a certain flavor of the optimal design problem, based on the Bayesian D-optimality criterion, and suggest an analytically tractable model for D-optimal designs in Bayesian linear inverse problems over Hilbert spaces. We demonstrate that measurement clusterization is a generic property of D-optimal designs, and prove that correlated noise between measurements mitigates clusterization. We also give a full characterization of D-optimal designs under our model: We prove that D-optimal designs uniformly reduce uncertainty in a select subset of prior covariance eigenvectors. Finally, we show how measurement clusterization is a consequence of the characterization mentioned above and the pigeonhole principle.