  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.
