We introduce a new method to jointly reduce the dimension of the input and output space of a high-dimensional function. Choosing a reduced input subspace influences which output subspace is relevant and vice versa. Conventional methods focus on reducing either the input or output space, even though both are often reduced simultaneously in practice. Our coupled approach naturally supports goal-oriented dimension reduction, where either an input or output quantity of interest is prescribed. We consider, in particular, goal-oriented sensor placement and goal-oriented sensitivity analysis, which can be viewed as dimension reduction where the most important output or, respectively, input components are chosen. Both applications present difficult combinatorial optimization problems with expensive objectives such as the expected information gain and Sobol indices. By optimizing gradient-based bounds, we can determine the most informative sensors and most sensitive parameters as the largest diagonal entries of some diagnostic matrices, thus bypassing the combinatorial optimization and objective evaluation.

翻译：本文提出一种联合降低高维函数输入与输出空间维度的新方法。选择降维后的输入子空间会影响相关输出子空间的选择，反之亦然。传统方法通常仅关注输入或输出空间的降维，然而实际应用中二者常需同时处理。我们的耦合方法天然支持目标导向的降维，可针对预设的输入或输出目标量进行优化。我们特别研究了目标导向的传感器布置与目标导向的敏感性分析——这两类问题可分别视为选择最重要输出分量或最重要输入分量的降维过程。两种应用均涉及目标函数计算昂贵（如期望信息增益与Sobol指数）的复杂组合优化问题。通过优化基于梯度的边界，我们可以将最具信息量的传感器与最敏感参数识别为特定诊断矩阵的最大对角元，从而规避组合优化与目标函数求值过程。