Statistical Barriers to Affine-equivariant Estimation

• URL: http://arxiv.org/abs/2310.10758v1
• Date: Mon, 16 Oct 2023 18:42:00 GMT
• Title: Statistical Barriers to Affine-equivariant Estimation
• Authors: Zihao Chen, Yeshwanth Cherapanamjeri
• Abstract summary: We investigate the quantitative performance of affine-equivariant estimators for robust mean estimation. We find that classical estimators are either quantitatively sub-optimal or lack any quantitative guarantees. We construct a new affine-equivariant estimator which nearly matches our lower bound.
• Score: 10.077727846124633
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We investigate the quantitative performance of affine-equivariant estimators for robust mean estimation. As a natural stability requirement, the construction of such affine-equivariant estimators has been extensively studied in the statistics literature. We quantitatively evaluate these estimators under two outlier models which have been the subject of much recent work: the heavy-tailed and adversarial corruption settings. We establish lower bounds which show that affine-equivariance induces a strict degradation in recovery error with quantitative rates degrading by a factor of $\sqrt{d}$ in both settings. We find that classical estimators such as the Tukey median (Tukey '75) and Stahel-Donoho estimator (Stahel '81 and Donoho '82) are either quantitatively sub-optimal even within the class of affine-equivariant estimators or lack any quantitative guarantees. On the other hand, recent estimators with strong quantitative guarantees are not affine-equivariant or require additional distributional assumptions to achieve it. We remedy this by constructing a new affine-equivariant estimator which nearly matches our lower bound. Our estimator is based on a novel notion of a high-dimensional median which may be of independent interest. Notably, our results are applicable more broadly to any estimator whose performance is evaluated in the Mahalanobis norm which, for affine-equivariant estimators, corresponds to an evaluation in Euclidean norm on isotropic distributions.

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