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.

翻译：多输出高斯过程回归已成为不确定性量化领域的重要工具，用于构建计算成本高昂的仿真器的替代模型，并广泛应用于多任务机器学习等其他领域。本文系统性地提出了张量变量高斯过程回归方法，该方法适用于任意维度的输出，其中协方差矩阵可采用克罗内克积结构。我们论证了针对二维输出问题的两种常见方法——外积仿真器与并行部分仿真器——均为张量变量高斯过程回归的特例，因此可扩展至更高维输出场景。聚焦于矩阵输出的重要特例，我们深入研究了这两种方法的相对性能差异。其核心区别在于外积仿真器所假设的附加依赖结构，我们通过两个案例研究（包括在时空流感仿真器中的应用）论证了该结构何时具有优势。