#### Quantum Entropy Scoring for Fast Robust Mean Estimation and Improved Outlier Detection

Part of Advances in Neural Information Processing Systems 32 (NeurIPS 2019)

#### Authors

*Yihe Dong, Samuel Hopkins, Jerry Li*

#### Abstract

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 $\mu$ of a distribution on $\mathbb{R}^d$ given $n$ independent samples, an $\epsilon$-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 $\tilde{O}(nd)$ in all parameters, improving on the previous fastest running time $\tilde{O}(\min(nd/\e^6, nd^2))$.
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.