Optimal Sensor Placement in Gaussian Processes via Column Subset Selection

Abstract

Gaussian process regression uses data measured at sensor locations to reconstruct a spatially dependent function with quantified uncertainty. However, if only a limited number of sensors can be deployed, it is important to determine how to optimally place the sensors to minimize a measure of the uncertainty in the reconstruction. We consider the Bayesian D-optimal criterion to determine the optimal sensor locations by choosing sensors from a candidate set of sensors. Since this is an NP-hard problem, our approach models sensor placement as a column subset selection problem (CSSP) on the covariance matrix, computed using the kernel function on the candidate sensor points. We propose an algorithm that uses the Golub-Klema-Stewart framework (GKS) to select sensors and provide an analysis of lower bounds on the D-optimality of these sensor placements. To reduce the computational cost in the GKS step, we propose and analyze algorithms for the D-optimal sensor placements using Nystr\"om approximations on the covariance matrix. Moreover, we propose several algorithms that select sensors via Nystr\"om approximation of the covariance matrix, utilizing the randomized Nystr\"om approximation, random pivoted Cholesky and greedy pivoted Cholesky. We demonstrate the performance of our method on two applications: thin liquid film dynamics and sea surface temperature.