Generalized bounds for active subspaces
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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.
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