Tensor-variate Gaussian process regression for efficient emulation of complex systems: comparing regressor and covariance structures in outer product and parallel partial emulators

TL;DR

This paper develops tensor-variate Gaussian process regression for multi-dimensional outputs, unifying existing approaches and analyzing their performance in complex system emulation, with applications to spatial-temporal models.

Contribution

It introduces a unified tensor-variate Gaussian process framework that generalizes outer product and parallel partial emulators to higher dimensions.

Findings

OPE and PPE are special cases of TvGP regression.

The additional dependence structure in OPE can be advantageous.

Application to a spatial-temporal influenza simulator demonstrates practical benefits.

Abstract

Multi-output Gaussian process regression has become an important tool in uncertainty quantification, for building emulators of computationally expensive simulators, and other areas such as multi-task machine learning. We present a holistic development of tensor-variate Gaussian process (TvGP) regression, appropriate for arbitrary dimensional outputs where a Kronecker product structure is appropriate for the covariance. We show how two common approaches to problems with two-dimensional output, outer product emulators (OPE) and parallel partial emulators (PPE), are special cases of TvGP regression and hence can be extended to higher output dimensions. Focusing on the important special case of matrix output, we investigate the relative performance of these two approaches. The key distinction is the additional dependence structure assumed by the OPE, and we demonstrate when this is advantageous through two case studies, including application to a spatial-temporal influenza simulator.