IMPROVED COVARIANCE ESTIMATION: OPTIMAL ROBUSTNESS AND SUB-GAUSSIAN GUARANTEES UNDER HEAVY TAILS

• 成果类型: Article
• 署名作者: Oliveira, Roberto i.; Rico, Zoraida f.
• 署名单位: Instituto Nacional de Matematica Pura e Aplicada (IMPA); Columbia University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/24-AOS2407
• 发表日期: 2024
• 页码: 1953-1977
• 关键词: Matrices; bounds

摘要

We present an estimator of the covariance matrix Sigma of random ddimensional vector from an i.i.d. sample of size n. Our sole assumption is that this vector satisfies a bounded L-p - L-2 moment assumption over its onedimensional marginals, for some p > 4. Given this, we show that E can be estimated from the sample with the same high-probability error rates that the sample covariance matrix achieves in the case of Gaussian data. This holds even though we allow for very general distributions that may not have moments of order > p. Moreover, our estimator can be made to be optimally robust to adversarial contamination. This result improves the recent contributions by Mendelson and Zhivotovskiy and Catoni and Giulini, and matches parallel work by Abdalla and Zhivotovskiy (the exact relationship with this last work is described in the paper).