用于高效仿真复杂系统的张量变量高斯过程回归：比较外积仿真器与并行偏仿真器中的回归器和协方差结构
Tensor-variate Gaussian process regression for efficient emulation of complex systems: comparing regressor and covariance structures in outer product and parallel partial emulators
AI总结：
本文系统发展张量变量高斯过程回归，统一外积与并行偏仿真器，并通过案例比较二者在矩阵输出下的协方差依赖结构与性能。
摘要：
多输出高斯过程回归已成为不确定性量化中的重要工具，用于构建计算昂贵仿真器的仿真模型，以及多任务机器学习等其他领域。我们提出了张量变量高斯过程（TvGP）回归的整体性发展，该方法适用于任意维度输出，且其协方差适合采用 Kronecker 积结构。我们展示了针对二维输出问题的两种常用方法——外积仿真器（OPE）和并行偏仿真器（PPE）——如何成为 TvGP 回归的特例，并因此可扩展到更高输出维度。围绕矩阵输出这一重要特例，我们研究了这两种方法的相对性能。关键区别在于 OPE 所假设的额外依赖结构；我们通过两个案例研究（包括在一个时空流感仿真器中的应用）证明了这种结构何时具有优势。
英文摘要：
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.
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