This paper proposes several approaches as baselines to compute a shared active subspace for multivariate vector-valued functions. The goal is to minimize the deviation between the function evaluations on the original space and those on the reconstructed one. This is done either by manipulating the gradients or the symmetric positive (semi-)definite (SPD) matrices computed from the gradients of each component function so as to get a single structure common to all component functions. These approaches can be applied to any data irrespective of the underlying distribution unlike the existing vector-valued approach that is constrained to a normal distribution. We test the effectiveness of these methods on five optimization problems. The experiments show that, in general, the SPD-level methods are superior to the gradient-level ones, and are close to the vector-valued approach in the case of a normal distribution. Interestingly, in most cases it suffices to take the sum of the SPD matrices to identify the best shared active subspace.

翻译：本文提出了几种基线方法，用于计算多元向量值函数的共享活动子空间。其目标是最小化原始空间上的函数评估与重构空间上的函数评估之间的偏差。这可通过操控每个分量函数的梯度，或由其梯度计算的对称正（半）定矩阵（SPD矩阵）来实现，从而获得所有分量函数共有的单一结构。与现有受限于正态分布的向量值方法不同，这些方法可适用于任意分布的数据。我们在五个优化问题上测试了这些方法的有效性。实验表明，总体而言，基于SPD矩阵的方法优于基于梯度的方法，且在正态分布情况下接近向量值方法的性能。有趣的是，在大多数情况下，仅需取SPD矩阵之和即可识别出最优共享活动子空间。