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

TL;DR

This paper introduces a quantum entropy-based outlier scoring method called QUE-scoring, which improves robust mean estimation accuracy and efficiency, and enhances outlier detection performance in high-dimensional data.

Contribution

The paper presents a novel QUE-scoring method based on quantum entropy regularization, achieving optimal error rates and faster algorithms for robust mean estimation and outlier detection.

Findings

First algorithm with optimal error rates for robust mean estimation

Nearly-linear time complexity in all parameters for the proposed method

QUE-scoring often outperforms previous outlier detection algorithms in experiments

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 $\varepsilon$-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 .