Year: 2024
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Gamarnik, David, Smedira, Devin
Institution: Massachusetts Institute of Technology
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4391833494