Preprint: Generalized bounds for active subspaces
I am very happy to announce that Jonas Wallin (Department of Statistics, Lund University), Barbara Wohlmtuh (Chair for Numerical Mathematics, TUM), and I put a new article on theory for active subspaces (ASM) on arXiv. ASM is already known to be applicable to settings involving probability distributions with compact support or of Gaussian-type. We investigate ASM in the context of distributions with exponential tails and were able to show that existing bounds are not valid anymore due to arbitrarily large Poincaré constants. Also, we propose a way for getting weaker, or generalized, bounds that, however, result in a lower order of the error bound. Indeed, we show how to balance the size the Poincaré constant and the order of the error. At the end, we suggest an open problem to the community which aims at generalizing the applicability of ASM to a larger class of distributions, i.e., multivariate generalized hyperbolics. I want to thank both of my co-authors for their feedback and assistance in developing the structure of this manuscript and for the valuable discussions we had.
Abstract. The active subspace method, as a dimension reduction technique, can substantially reduce computational costs and is thus attractive for high-dimensional computer simulations. The theory provides upper bounds for the mean square error of a given function of interest and a low-dimensional approximation of it. Derivations are based on probabilistic Poincaré inequalities which strongly depend on an underlying probability distribution that weights sensitivities of the investigated function. It is not this original distribution that is crucial for final error bounds, but a conditional distribution, conditioned on a so-called active variable, that naturally arises in the context. Existing literature does not take this aspect into account, is thus missing important details when it comes to distributions with, for example, exponential tails, and, as a consequence, does not cover such distributions theoretically. Here, we consider scenarios in which traditional estimates are not valid anymore due to an arbitrary large Poincaré constant. Additionally, we propose a framework that allows to get weaker, or generalized, estimates and that enables the practitioner to control the trade-off between the size of the Poincaré type constant and a weaker order of the final error bound. In particular, we investigate independently exponentially distributed random variables in 2 and n dimensions and give explicit expressions for involved constants, also showing the dependence on the dimension of the problem. Finally, we formulate an open problem to the community that aims for extending the class of distributions applicable to the active subspace method as we regard this as an opportunity to enlarge its usability.