We introduce a new method to jointly reduce the dimension of the input and output space of a function between high-dimensional spaces. 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 influential parameters as the largest diagonal entries of some diagnostic matrices, thus bypassing the combinatorial optimization and objective evaluation.

翻译：本文提出一种新方法，用于联合降维高维空间之间函数的输入与输出空间。选择降维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场景，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。两类应用均涉及复杂的组合优化问题，其目标函数（如期望信息增益与Sobol指数）的计算代价高昂。通过优化基于梯度的边界条件，我们可以将最具信息量的传感器和最具影响力的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。