信息论视角下的基数估计极限：费舍尔与香农的交汇
Information Theoretic Limits of Cardinality Estimation: Fisher Meets Shannon
AI总结：
本文提出FiSh数度量数据草图效率，证明Fishmonger草图在基数估计中达到理论最优，且H₀/I₀为可合并草图的最优FiSh数。
摘要：
估计大规模多重集合的基数（即不同元素的数量）是流式计算和草图技术中的经典问题。本文研究了草图的空间复杂度与其估计误差之间的内在权衡。我们定义了一个新的数据草图效率度量标准，称为费舍尔-香农（FiSh）数H/I。它捕捉了草图的极限香农熵（H）与归一化的费舍尔信息（I）之间的张力，后者表征了统计上高效且渐近无偏估计量的方差。引入FiSh数的目的是为了建立数学工具，以论证精确最优性，而非渐近最优性，直到较大的常数因子。我们的结果如下。[1] 我们证明了所有基数为q的Flajolet和Martin的PCSA草图变体的FiSh数为H₀/I₀≈1.98016，而所有基数为q的HyperLogLog变体的FiSh数劣于H₀/I₀，但随着q趋近于无穷大时，它们会趋近于H₀/I₀。这里H₀、I₀是精确定义的常数。[2] 我们描述了一种名为Fishmonger的草图，其基于一种平滑、熵压缩的PCSA变体，具有不同的估计函数。Fishmonger处理一个[U]的多重集合，使得在所有时间点，以高概率其空间为(1+o(1))(H₀/I₀)m≈1.98m位，其标准误差为1/√m。例如，要实现1%的标准误差，需要约19,800位，或约2.42千字节。[3] 最后，我们提供了间接证据表明H₀/I₀是可合并草图的基数估计的最优FiSh数。我们定义了一个自然的可合并草图子集，称为线性化草图，并证明该类中没有任何成员能超越H₀/I₀。实际上，流行的可合并草图也是线性化的。
英文摘要：
Estimating the cardinality (number of distinct elements) of a large multiset is a classic problem in streaming and sketching. In this paper we study the intrinsic tradeoff between the space complexity of the sketch and its estimation error. We define a new measure of efficiency for data sketches called the Fisher-Shannon (FiSh) number $\mathcal{H}/\mathcal{I}$. It captures the tension between the limiting Shannon entropy ($\mathcal{H}$) of the sketch and its normalized Fisher information ($\mathcal{I}$) that characterizes the variance of a statistically efficient, asymptotically unbiased estimator. Our aim in introducing the FiSh-number is to build the mathematical machinery necessary to argue for precise optimality, rather than asymptotic optimality, up to large constant factors. Our results are as follows. [1] We prove that all base-$q$ variants of Flajolet and Martin's PCSA sketch have FiSh-number $H_0/I_0 \approx 1.98016$ and that every base-$q$ variant of HyperLogLog has FiSh-number worse than $H_0/I_0$, but that they tend to $H_0/I_0$ in the limit as $q\rightarrow \infty$. Here $H_0,I_0$ are precisely defined constants. [2] We describe a sketch called Fishmonger that is based on a smoothed, entropy-compressed variant of PCSA with a different estimator function. Fishmonger processes a multiset of $[U]$ such that at all times, w.h.p., its space is $(1+o(1))(H_0/I_0)m \approx 1.98m$ bits and its standard error is $1/\sqrt{m}$. For example, to achieve a 1% standard error, one needs a little more than 19,800 bits, or $\approx 2.42$ kilobytes. [3] Finally, we give circumstantial evidence that $H_0/I_0$ is the optimum FiSh-number of mergeable sketches for Cardinality Estimation. We define a natural subset of mergeable sketches called linearizable sketches and prove that no member of this class can beat $H_0/I_0$. The popular mergeable sketches are, in fact, also linearizable.
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