# Integrating High-Dimensional Functions Deterministically

CoRR（2024）

Abstract

We design a Quasi-Polynomial time deterministic approximation algorithm for
computing the integral of a multi-dimensional separable function, supported by
some underlying hyper-graph structure, appropriately defined. Equivalently, our
integral is the partition function of a graphical model with continuous
potentials. While randomized algorithms for high-dimensional integration are
widely known, deterministic counterparts generally do not exist. We use the
correlation decay method applied to the Riemann sum of the function to produce
our algorithm. For our method to work, we require that the domain is bounded
and the hyper-edge potentials are positive and bounded on the domain. We
further assume that upper and lower bounds on the potentials separated by a
multiplicative factor of 1 + O(1/Δ^2), where Δ is the maximum
degree of the graph. When Δ = 3, our method works provided the upper and
lower bounds are separated by a factor of at most 1.0479. To the best of our
knowledge, our algorithm is the first deterministic algorithm for
high-dimensional integration of a continuous function, apart from the case of
trivial product form distributions.

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