基于量子熵的快速稳健均值估计及改进的离群点检测
Quantum Entropy Scoring for Fast Robust Mean Estimation and Improved Outlier Detection
摘要：
我们研究了高维稳健统计中的两个问题：\emph{robust mean estimation}和\emph{outlier detection}。在鲁棒均值估计中，目标是估计给定$n$独立样本的$\mathbb{R}^d$上的分布的均值$μ$，其中$\varepsilon$部分已被恶意攻击者破坏。在离群值检测中，目标是为数据集的每个元素分配一个\emph{outlier score}，以便为更可能是离群值的元素分配更高的分数。这两个问题的算法是基于一个新的离群值评分方法，我们称之为QUE评分的基础上\emph{quantum entropy regularization}。对于鲁棒均值估计，这产生了第一个在所有参数中具有最佳错误率和接近线性运行时间$\widetilde{O}(nd)$的算法，改进了先前的最快运行时间$\widetilde{O}(\min(nd/\varepsilon^6, nd^2))$。对于离群值检测，我们通过对合成数据和真实数据的大量实验来评估QUE评分的性能，并证明它通常比以前提出的算法表现得更好。这些实验的代码可在https://github.com/twistedcubic/que-outlier-detection上获得。
英文摘要：
We study two problems in high-dimensional robust statistics: \emph{robust mean estimation} and \emph{outlier detection}. In robust mean estimation the goal is to estimate the mean $μ$ of a distribution on $\mathbb{R}^d$ given $n$ independent samples, an $\varepsilon$-fraction of which have been corrupted by a malicious adversary. In outlier detection the goal is to assign an \emph{outlier score} to each element of a data set such that elements more likely to be outliers are assigned higher scores. Our algorithms for both problems are based on a new outlier scoring method we call QUE-scoring based on \emph{quantum entropy regularization}. For robust mean estimation, this yields the first algorithm with optimal error rates and nearly-linear running time $\widetilde{O}(nd)$ in all parameters, improving on the previous fastest running time $\widetilde{O}(\min(nd/\varepsilon^6, nd^2))$. For outlier detection, we evaluate the performance of QUE-scoring via extensive experiments on synthetic and real data, and demonstrate that it often performs better than previously proposed algorithms. Code for these experiments is available at https://github.com/twistedcubic/que-outlier-detection .
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