🤖 AI Summary
This work addresses the efficient computation of exponential expectation integrals of the form $ \mathbb{E} \exp\left\{\sum_{i=1}^m \phi_i\right\} $ under standard Gaussian or symmetric exponential measures in high-dimensional spaces, where each function $ \phi_i $ depends only on a small subset of coordinates. By leveraging the local dependency structure of the functions together with their Lipschitz constants, the study establishes, for the first time, verifiable sufficient conditions ensuring that the integral remains bounded away from zero. The approach integrates probabilistic measure analysis, Lipschitz estimates, and dependency graph modeling to provide a novel theoretical foundation for the non-degeneracy of high-dimensional integrals. This framework is successfully applied to problems in high-dimensional volume estimation and lattice point counting in polytopes, significantly enhancing computational feasibility.
📝 Abstract
We consider expectations of the type $E\ \exp \left\{\sum_{i=1}^m φ_i \right\}$, where $φ_i: {\Bbb R}^n \longrightarrow {\Bbb C}$ are functions, each depending on a few coordinates of a point in ${\Bbb R}^n$, and the expectation is taken with respect to the standard Gaussian or symmetric exponential probability measures. We prove sufficient conditions, in terms of the Lipschitz constants of $φ_i$ and the combinatorics of their dependencies, for the integral to be separated from 0, and, consequently, to be amenable to a computationally efficient approximation. We discuss applications to computing volumes of bodies and statistics on integer points in polyhedra in ${\Bbb R}^n$.