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 $1 + O(1/\Delta^2)$, where $\Delta$ is the maximum degree of the graph. When $\Delta = 3$, 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.

翻译：我们设计了一种拟多项式时间确定性近似算法，用于计算具有适当定义超图结构支撑的多维可分离函数积分。等价地，该积分对应连续势能图模型的配分函数。尽管高维积分的随机算法已广为人知，但确定性算法通常不存在。我们应用相关衰减方法处理函数的黎曼和，从而构造出该算法。为保证方法有效性，要求函数定义域有界且超边势能在定义域内恒正且有界。进一步假设势能上下界之比不超过 $1 + O(1/\Delta^2)$，其中 $\Delta$ 为图的最大度。当 $\Delta = 3$ 时，该比值需不超过 $1.0479$。据我们所知，除平凡乘积型分布情形外，本算法是首个针对连续函数高维积分的确定性算法。