ROBUST COVARIANCE ESTIMATION UNDER L4 - L2 NORM EQUIVALENCE

• 成果类型: Article
• 署名作者: Mendelson, Shahar; Zhivotovskiy, Nikita
• 署名单位: Australian National University; HSE University (National Research University Higher School of Economics)
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/19-AOS1862
• 发表日期: 2020
• 页码: 1648-1664
• 关键词: matrix; bounds

摘要

Let X be a centered random vector taking values in R-d and let Sigma = E (X circle times X) be its covariance matrix. We show that if X satisfies an L-4 - L-2 norm equivalence (sometimes referred to as the bounded kurtosis assumption), there is a covariance estimator (Sigma) over cap that exhibits almost the same performance one would expect had X been a Gaussian vector. The procedure also improves the current state-of-the-art regarding high probability bounds in the sub-Gaussian case (sharp results were only known in expectation or with constant probability). In both scenarios the new bounds do not depend explicitly on the dimension d, but rather on the effective rank of the covariance matrix Sigma.