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.

在这篇研究论文中，我们引入了一种新的方法来同时降低输入和输出空间的维度。我们的联合方法支持目标导向的维度减少，其中输入或输出的感兴趣数量被预设。我们特别考虑了目标导向的传感器放置和目标导向的灵敏度分析，可以视为选择最重要的输出或输入组件的维度减少。通过优化梯度边界，我们可以确定最具信息量的传感器和最敏感的参数，从而避免组合优化和目标评估的困难。