论文标题
超收缩符合随机凸壳：随机多元库的分析
Hypercontractivity Meets Random Convex Hulls: Analysis of Randomized Multivariate Cubatures
论文作者
论文摘要
鉴于概率度量$ \ MATHCAL {X} $和矢量值函数$φ$的概率度量$μ$，一个常见的问题是在$ \ Mathcal {x} $上构建一个离散的概率度量，以便在$φ$下的这两个概率指标的推动力是相同的。这种结构是数值集成方法的核心，这些方法以各种名称（例如正交，立方体或重组）运行。一种自然的方法是将$μ$的样品点从$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $φ$的平均值中。在这里，当$φ$通过使用所谓的超收缩率表现出分级结构时，我们分析了这种方法的计算复杂性。所得定理不仅涵盖了多元多项式的经典群组案例，而且还涵盖了路径空间的集成以及用于产品测量的内核正交。
Given a probability measure $μ$ on a set $\mathcal{X}$ and a vector-valued function $φ$, a common problem is to construct a discrete probability measure on $\mathcal{X}$ such that the push-forward of these two probability measures under $φ$ is the same. This construction is at the heart of numerical integration methods that run under various names such as quadrature, cubature, or recombination. A natural approach is to sample points from $μ$ until their convex hull of their image under $φ$ includes the mean of $φ$. Here we analyze the computational complexity of this approach when $φ$ exhibits a graded structure by using so-called hypercontractivity. The resulting theorem not only covers the classical cubature case of multivariate polynomials, but also integration on pathspace, as well as kernel quadrature for product measures.
