论文标题
基于广义最小二乘的复合空间蒙特卡洛整合
Composite Spatial Monte Carlo Integration Based on Generalized Least Squares
论文作者
论文摘要
尽管在各种应用中对ISING模型的期望进行评估至关重要，但由于可怕的多个求和，这主要是不可行的。空间蒙特卡洛整合（SMCI）是基于抽样的近似。它可以为这种棘手的期望提供高临界性的估计。为了评估特定区域（称为目标区域）变量函数的期望，SMCI考虑了一个较大的区域，该区域包含目标区域（称为SUM区域）。在SMCI中，精确执行了SUM区域中变量的多个求和，并且在外部区域中，通过采样近似（例如标准的蒙特卡洛集成）来评估外部区域。可以确保SMCI估计量的准确性随着SUM区域的大小增加而单调地提高。但是，偶然的膨胀区域可能会导致组合爆炸。因此，我们希望在没有这种扩展的情况下提高准确性。在本文中，基于广义最小二乘理论（GL），通过组合多个SMCI估计器提出了一种新的有效方法。在理论和数值上证明了所提出方法的有效性。结果表明，所提出的方法可以在反iSing问题（或Boltzmann机器学习）中有效。
Although evaluation of the expectations on the Ising model is essential in various applications, it is mostly infeasible because of intractable multiple summations. Spatial Monte Carlo integration (SMCI) is a sampling-based approximation. It can provide high-accuracy estimations for such intractable expectations. To evaluate the expectation of a function of variables in a specific region (called target region), SMCI considers a larger region containing the target region (called sum region). In SMCI, the multiple summation for the variables in the sum region is precisely executed, and that in the outer region is evaluated by the sampling approximation such as the standard Monte Carlo integration. It is guaranteed that the accuracy of the SMCI estimator improves monotonically as the size of the sum region increases. However, a haphazard expansion of the sum region could cause a combinatorial explosion. Therefore, we hope to improve the accuracy without such an expansion. In this paper, based on the theory of generalized least squares (GLS), a new effective method is proposed by combining multiple SMCI estimators. The validity of the proposed method is demonstrated theoretically and numerically. The results indicate that the proposed method can be effective in the inverse Ising problem (or Boltzmann machine learning).
