Revised preprint: Generalized bounds for active subspaces
Jonas Wallin, Barbara Wohlmuth, and I put a revised version of our article Generalized bounds for active subspaces in the arXiv. The main changes consist of a formalization of our results to a theorem/proof style, the consideration of a particular supremum (more below), and a revision of the section on future work with MGH distributions (multivariate generalized hyperbolics).
In the former version, our counterexample to existing theoretical results considered an arbitrary orthogonal transformation of input variables that, however, was used before as a particular defined transformation. Since related quantities appear in error bounds, we now consider the supremum of the related quantities over the set of all orthogonal matrices which makes it valid for us to keep regard arbitrary transformations. In fact, we should justify why it is enough in our case to consider rotations, a subset of orthogonal transformations, only.
Finally, I want to thank both of my co-authors for their feedback and assistance in revising this manuscript.
Abstract. In this article, we consider scenarios in which traditional estimates for the active subspace method based on probabilistic Poincaré inequalities are not valid due to unbounded Poincaré constants. Consequently, we propose a framework that allows to derive generalized estimates in the sense that it enables to control the trade-off between the size of the Poincaré constant and a weaker order of the final error bound. In particular, we investigate independently exponentially distributed random variables in dimension two or larger and give explicit expressions for corresponding Poincaré constants showing their dependence on the dimension of the problem. Finally, we suggest possibilities for future work that aim for extending the class of distributions applicable to the active subspace method as we regard this as an opportunity to enlarge its usability.