多目标加权抽样
Multi-Objective Weighted Sampling
摘要：
{ \em多目标样本}是大型数据集的功能强大且用途广泛的汇总。对于一组密钥$x\in X$和相关联的值$f_x \geq 0$，相对于$f$采取的加权样本允许我们对于密钥的任何子集$H$近似{ \em分段-和统计量} $\text{Sum}(f;H) = \text{sum}_{x\in H} f_x$，具有取决于样本大小和$H$的相对权重的严格保证的质量。然而，当估计$g\not=f$的$\text{Sum}(g;H)$时，质量保证将丢失。关于一组函数$F$的多目标样本为每个$f\in F$提供与专用加权样本相同的统计保证，同时最小化汇总大小。我们分析了多目标样本的属性，并提出了估计和优化的采样方案和元算法，同时展示了两个重要的应用领域。第一种是键值数据集，其中应用于值的不同函数$f\in F$对应于不同的统计数据，如矩、阈值、上限和总和。多目标样本允许我们近似$F$中的所有统计数据。第二种是度量空间，其中键是点，并且每个$f\in F$由一组点$C$定义，其中$f_x$是$x$乘$C$的服务成本，并且$\text{Sum}(f;X)$模型$C$的中心性或聚类成本。多目标样本使我们能够估计每个$f\in F$的成本。在这些领域中，多目标样本通常是小尺寸的，可以有效地构建，并且能够进行可扩展的估计和优化。我们的目标是促进这一强大技术的进一步应用。
英文摘要：
{\em Multi-objective samples} are powerful and versatile summaries of large data sets. For a set of keys $x\in X$ and associated values $f_x \geq 0$, a weighted sample taken with respect to $f$ allows us to approximate {\em segment-sum statistics} $\text{Sum}(f;H) = \text{sum}_{x\in H} f_x$, for any subset $H$ of the keys, with statistically-guaranteed quality that depends on sample size and the relative weight of $H$. When estimating $\text{Sum}(g;H)$ for $g\not=f$, however, quality guarantees are lost. A multi-objective sample with respect to a set of functions $F$ provides for each $f\in F$ the same statistical guarantees as a dedicated weighted sample while minimizing the summary size. We analyze properties of multi-objective samples and present sampling schemes and meta-algortithms for estimation and optimization while showcasing two important application domains. The first are key-value data sets, where different functions $f\in F$ applied to the values correspond to different statistics such as moments, thresholds, capping, and sum. A multi-objective sample allows us to approximate all statistics in $F$. The second is metric spaces, where keys are points, and each $f\in F$ is defined by a set of points $C$ with $f_x$ being the service cost of $x$ by $C$, and $\text{Sum}(f;X)$ models centrality or clustering cost of $C$. A multi-objective sample allows us to estimate costs for each $f\in F$. In these domains, multi-objective samples are often of small size, are efficiently to construct, and enable scalable estimation and optimization. We aim here to facilitate further applications of this powerful technique.
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