Quantum Entropy Scoring for Fast Robust Mean Estimation and Improved Outlier Detection
This work addresses the problem of efficient and accurate robust statistical analysis for data scientists, offering significant speed improvements and competitive results, though it is incremental in building on existing outlier scoring methods.
The paper tackles robust mean estimation and outlier detection in high-dimensional statistics by introducing QUE-scoring based on quantum entropy regularization, achieving optimal error rates with nearly-linear runtime O~(nd) for robust mean estimation and showing improved performance in outlier detection experiments.
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 .