High dimensional integration is essential to many areas of science, ranging from particle physics to Bayesian inference. Approximating these integrals is hard, due in part to the difficulty of locating and sampling from regions of the integration domain that make significant contributions to the overall integral. Here, we present a new algorithm called Tree Quadrature (TQ) that separates this sampling problem from the problem of using those samples to produce an approximation of the integral. TQ places no qualifications on how the samples provided to it are obtained, allowing it to use state-of-the-art sampling algorithms that are largely ignored by existing integration algorithms. Given a set of samples, TQ constructs a surrogate model of the integrand in the form of a regression tree, with a structure optimised to maximise integral precision. The tree divides the integration domain into smaller containers, which are individually integrated and aggregated to estimate the overall integral. Any method can be used to integrate each individual container, so existing integration methods, like Bayesian Monte Carlo, can be combined with TQ to boost their performance. On a set of benchmark problems, we show that TQ provides accurate approximations to integrals in up to 15 dimensions; and in dimensions 4 and above, it outperforms simple Monte Carlo and the popular Vegas method.
Fast Approximation and Estimation Bounds of Kernel Quadrature for Infinitely Wide Models
RIKEN AIP（日本理化学研究所革新智能综合研究中心）
无限宽模型是特征图的加权积分$\int φ(x,v) d μ(v)$。该模型擅长处理无限数量的特征，因此已被用于深度学习的理论研究。核求积是一种基于核的数值积分方案，用于快速逼近期望值$\int f(x) d p(x)$。在这项研究中，关于重量$μ$作为一个有符号（或复杂/向量值）分布的参数，我们开发的一般核求积（GKQ）的参数分布。所提出的方法可以实现一个快速的近似速度$O(e^{-p})$与参数数$p$，这是比传统的巴伦的速度，和一个快速的估计速度$\widetilde{O}(1/n)$与样本量$n$。因此，我们得到了一个新的基于范数的复杂性度量的无限宽模型。由于GKQ隐式地进行经验风险最小化，因此我们可以理解复杂性度量也反映了梯度学习设置中的泛化性能。
Graduate School of Information Science and Technology, The University of Tokyo（东京大学信息科学与技术研究生院）
核方法是分析结构化数据（如序列、树和图）的一种潜在方法，然而，无序树尚未得到广泛研究。Kimura et al.（2011）提出了一种基于子路径的无序树的核函数，子路径是树的垂直子结构，负责其中的层次信息。他们的内核在精度和速度方面表现出实际上很好的性能;然而，线性时间计算在理论上不能保证，不像Vishwanathan和Smola（2003）提出的其他无序树内核。在本文中，我们提出了一个理论上保证线性时间内核计算算法，实际上是快速的，我们提出了一个有效的预测算法，其运行时间只取决于输入树的大小。实验结果表明，本文提出的算法在实际应用中是非常有效的.
