  Multivariate functions encountered in high-dimensional uncertainty
quantification problems often vary most strongly along a few dominant
directions in the input parameter space. We propose a gradient-based method for
detecting these directions and using them to construct ridge approximations of
such functions, in the case where the functions are vector-valued (e.g., taking
values in $\mathbb{R}^n$). The methodology consists of minimizing an upper
bound on the approximation error, obtained by subspace Poincar\'e inequalities.
We provide a thorough mathematical analysis in the case where the parameter
space is equipped with a Gaussian probability measure. The resulting method
generalizes the notion of active subspaces associated with scalar-valued
functions. A numerical illustration shows that using gradients of the function
yields effective dimension reduction. We also show how the choice of norm on
the codomain of the function has an impact on the function's low-dimensional
approximation.
