We provide a novel statistical perspective on a classical problem at the intersection of computer science and information theory: recovering the empirical frequency of a symbol in a large discrete dataset using only a compressed representation, or sketch, obtained via random hashing. Departing from traditional algorithmic approaches, recent works have proposed Bayesian nonparametric (BNP) methods that can provide more informative frequency estimates by leveraging modeling assumptions about the distribution of the sketched data. In this paper, we propose a {\em smoothed-Bayesian} method, inspired by existing BNP approaches but designed in a frequentist framework to overcome the computational limitations of the BNP approaches when dealing with large-scale data from realistic distributions, including those with power-law tail behaviors. For sketches obtained with a single hash function, our approach is supported by rigorous frequentist properties, including unbiasedness and optimality under a squared error loss function within an intuitive class of linear estimators. For sketches with multiple hash functions, we introduce an approach based on \emph{multi-view} learning to construct computationally efficient frequency estimators. We validate our method on synthetic and real data, comparing its performance to that of existing alternatives.

翻译：本文针对计算机科学与信息论交叉领域的一个经典问题提出了新颖的统计学视角：仅利用通过随机哈希函数获得的压缩表示（即草图），从大规模离散数据集中恢复符号的经验频率。不同于传统算法方法，近期研究提出的贝叶斯非参数方法可通过利用草图数据分布的建模假设提供更具信息量的频率估计。本文提出一种"平滑贝叶斯"方法，该方法受现有贝叶斯非参数方法启发，但在频率学派框架下设计，旨在克服贝叶斯非参数方法处理大规模现实分布（包括具有幂律尾行为的分布）数据时的计算局限性。对于单哈希函数草图，我们的方法具备严格的频率学派性质，包括无偏性以及在平方误差损失函数下线性估计量直观类中的最优性。针对多哈希函数草图，我们引入基于"多视图"学习的方法来构建计算高效的频率估计量。通过在合成数据与真实数据上的实验，我们将本方法与现有替代方案的性能进行了比较验证。