论文标题
内核插值与连续体积采样
Kernel interpolation with continuous volume sampling
论文作者
论文摘要
内核方法中的一个基本任务是挑选节点和权重，以便通过位于节点上的内核转换的加权总和近似RKHS的给定函数。这是内核密度估计，内核正交或离散样品中插值的关键。此外，RKHSS提供了方便的数学和计算框架。我们介绍和分析连续体积采样（VS），即在（Deshpande＆Vempala，2006年）中引入的离散分布的连续对应物（用于选择节点位置）。我们的贡献是理论上的：我们证明了VS下的插值和正交的最佳界限。尽管使用临时节点构造已经存在一些特定的RKHS，但VS提供了适用于任何Mercer内核的边界，并取决于相关的集成运算符的光谱。我们强调的是，与以前依赖正规杠杆分数或确定点过程的随机方法不同，评估VS的PDF仅需要对内核的刻度评估。因此，VS自然适合MCMC采样器。
A fundamental task in kernel methods is to pick nodes and weights, so as to approximate a given function from an RKHS by the weighted sum of kernel translates located at the nodes. This is the crux of kernel density estimation, kernel quadrature, or interpolation from discrete samples. Furthermore, RKHSs offer a convenient mathematical and computational framework. We introduce and analyse continuous volume sampling (VS), the continuous counterpart -- for choosing node locations -- of a discrete distribution introduced in (Deshpande & Vempala, 2006). Our contribution is theoretical: we prove almost optimal bounds for interpolation and quadrature under VS. While similar bounds already exist for some specific RKHSs using ad-hoc node constructions, VS offers bounds that apply to any Mercer kernel and depend on the spectrum of the associated integration operator. We emphasize that, unlike previous randomized approaches that rely on regularized leverage scores or determinantal point processes, evaluating the pdf of VS only requires pointwise evaluations of the kernel. VS is thus naturally amenable to MCMC samplers.
