Optimal Sensor Placement in Gaussian Processes via Column Subset Selection
Jessie Chen, Hangjie Ji, Arvind K. Saibaba
TL;DR
This work addresses optimal sensor placement for Gaussian process regression when only a limited number of sensors can be deployed. It recasts the problem as a column subset selection task on the GP covariance matrix and develops a conceptual Golub-Klema-Stewart (GKS) framework to achieve near-optimal D-optimal designs, complemented by Nyström-based approximations (NysGKS) to scale to large candidate sets. The authors also introduce pivoted Cholesky variants and greedy baselines, providing theoretical bounds on D-optimality and eigenvalue behavior under Nyström approximations, and validate the approaches on thin liquid film dynamics and sea surface temperature data, showing favorable performance and uncertainty-aware reconstructions. Overall, the paper delivers scalable, mathematically grounded sensor-placement algorithms with competitive accuracy and robust uncertainty quantification for GP-based surrogate modeling. The methods are particularly appealing for large-scale geophysical and engineering applications where data collection is costly or impractical.
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 Nyström approximations on the covariance matrix. Moreover, we propose several algorithms that select sensors via Nyström approximation of the covariance matrix, utilizing the randomized Nyström 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.