In this work we take a new approach to constructing a universal sketch that focuses on a class of \emph{basis functions} $\{f_s(x)=1-\cos(sx)\mid s>0\}$, so that any $f$-moment can be estimated if $f$ can be expressed as a linear combination of basis functions. We construct and analyze the $\mathsf{SymmetricPoissonTower}$ sketch, which occupies $O(\epsilon^{-2}\log^2(nM\epsilon^{-1}))$ bits and is $\mathcal{F}$-universal for the function class $$\mathcal{F}= \left\{f(x)=cx^2+\int_0^\infty (1-\cos (xs))\,\nu(ds) \mid c\geq 0, \text{$\nu$ is a positive measure}\right\},$$ i.e., given any $f\in \mathcal{F}$, the new sketch $(1\pm\epsilon)$-estimates the $f$-moment with probability 2/3. The family $\mathcal{F}$ includes all the classic frequency moments ($f(z)=|z|^p$, $p\in [0,2]$) as well as a large family of nearly-periodic functions that cannot be estimated with $L_2$-heavy hitter machinery. This new approach to universality requires significantly less space in comparison to previous universal schemes and sheds new light on the full characterization of the class $\mathcal{T}$ of tractable functions.

翻译：在本研究中，我们采用一种新方法构建通用草图，其核心聚焦于一类\emph{基函数} $\{f_s(x)=1-\cos(sx)\mid s>0\}$，使得任何$f$矩均可在$f$可表示为基函数线性组合时进行估计。我们构建并分析了$\mathsf{SymmetricPoissonTower}$草图，该草图占用$O(\epsilon^{-2}\log^2(nM\epsilon^{-1}))$比特，且对函数类$$\mathcal{F}= \left\{f(x)=cx^2+\int_0^\infty (1-\cos (xs))\,\nu(ds) \mid c\geq 0, \text{$\nu$为正测度}\right\}$$具有$\mathcal{F}$-普适性，即对于任意$f\in \mathcal{F}$，新草图能以2/3概率实现$f$矩的$(1\pm\epsilon)$估计。函数族$\mathcal{F}$包含所有经典频率矩（$f(z)=|z|^p$, $p\in [0,2]$）以及一大类无法用$L_2$-频繁项技术估计的近似周期函数。相较于先前的通用方案，这种新的普适性构建方法所需空间显著减少，并为完全刻画可处理函数类$\mathcal{T}$的特征提供了新的理论视角。