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

We study two problems in high-dimensional robust statistics: robust mean estimation and outlier detection. In robust mean estimation the goal is to estimate the mean μ of a distribution on R^d given n independent samples, an ε-fraction of which have been corrupted by a malicious adversary. In outlier detection the goal is to assign an 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 quantum entropy regularization. For robust mean estimation, this yields the first algorithm with optimal error rates and nearly-linear running time Õ(nd) in all parameters, improving on the previous fastest running time Õ(min(nd/ε⁶, nd²)). 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 on Github (opens in new tab).