In the turnstile streaming model, a dynamic vector $\mathbf{x}=(\mathbf{x}_1,\ldots,\mathbf{x}_n)\in \mathbb{Z}^n$ is updated by a stream of entry-wise increments/decrements. Let $f\colon\mathbb{Z}\to \mathbb{R}_+$ be a symmetric function with $f(0)=0$. The \emph{$f$-moment} of $\mathbf{x}$ is defined to be $f(\mathbf{x}) := \sum_{v\in[n]}f(\mathbf{x}_v)$. We revisit the problem of constructing a \emph{universal sketch} that can estimate many different $f$-moments. Previous constructions of universal sketches rely on the technique of sampling with respect to the $L_0$-mass (uniform samples) or $L_2$-mass ($L_2$-heavy-hitters), whose universality comes from being able to evaluate the function $f$ over the samples. In this work we take a new approach to constructing a universal sketch that does not use \emph{any} explicit samples but relies on the \emph{harmonic structure} of the target function $f$. The new sketch ($\textsf{SymmetricPoissonTower}$) \emph{embraces} hash collisions instead of avoiding them, which saves multiple $\log n$ factors in space, e.g., when estimating all $L_p$-moments ($f(z) = |z|^p,p\in[0,2]$). For many nearly periodic functions, the new sketch is \emph{exponentially} more efficient than sampling-based methods. We conjecture that the $\textsf{SymmetricPoissonTower}$ sketch is \emph{the} universal sketch that can estimate every tractable function $f$.

翻译：在旋转门流模型中，动态向量 $\mathbf{x}=(\mathbf{x}_1,\ldots,\mathbf{x}_n)\in \mathbb{Z}^n$ 通过一系列逐项增减操作进行更新。令 $f\colon\mathbb{Z}\to \mathbb{R}_+$ 为满足 $f(0)=0$ 的对称函数。$\mathbf{x}$ 的 \emph{$f$-矩} 定义为 $f(\mathbf{x}) := \sum_{v\in[n]}f(\mathbf{x}_v)$。本文重新审视构建能够估计多种不同 $f$-矩的 \emph{通用草图} 的问题。现有的通用草图构造方法依赖于基于 $L_0$-质量（均匀采样）或 $L_2$-质量（$L_2$-重击者）的采样技术，其通用性源于能够对采样点计算函数 $f$。本文提出一种构建通用草图的新方法，该方法不依赖任何显式采样，而是基于目标函数 $f$ 的 \emph{调和结构}。新草图（$\textsf{SymmetricPoissonTower}$）\emph{主动利用}哈希碰撞而非避免碰撞，从而在空间复杂度上节省了多个 $\log n$ 因子，例如在估计所有 $L_p$-矩（$f(z) = |z|^p,p\in[0,2]$）时。对于许多近周期函数，新草图相比基于采样的方法具有 \emph{指数级} 的效率优势。我们推测 $\textsf{SymmetricPoissonTower}$ 草图是能够估计所有可处理函数 $f$ 的 \emph{终极} 通用草图。