Robust covariance estimation under $L_{4}-L_{2}$ norm equivalence

By Shahar Mendelson, Nikita Zhivotovskiy. Annals of Statistics, 2018

69 citations recorded by the source corpus.

Let $X$ be a centered random vector taking values in $\mathbb{R}^d$ and let $\Sigma= \mathbb{E}(X\otimes X)$ be its covariance matrix. We show that if $X$ satisfies an $L_4-L_2$ norm equivalence, there is a covariance estimator $\hat{\Sigma}$ that exhibits the optimal 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 subgaussian case (sharp results were only known in expectation or with constant probability). In both scenarios the new bound does not depend explicitly on the dimension $d$, but rather on the effective rank of the covariance matrix $\Sigma$.

Subfields: Artificial Intelligence, Computational Mechanics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e902068c3aa01222ae29832c6ef01ee4f4a032cb

Citations in the corpus

• Mean Estimation and Regression Under Heavy-Tailed Distributions: A Survey (2019, unknown): “An important example is the matrix operator norm, see Minsker [66], Catoni and Giulini [15], Mendelson and Zhivotovskiy [64].”
• Finite-sample Analysis of M-estimators using Self-concordance (2018, unknown): “The closest in this direction is the recent work [40] which establishes a high-probability bound in the operator norm, ‖Σ̂ − Σ‖ ≤ c‖Σ‖, holding with probability ≥ 1− δ when n ≥ C(κ)[r(Σ) + log(1/δ)], where C(κ) is a constant depending only on the kurtosis, and r(Σ) := Tr(Σ)/‖Σ‖ ≤ d is the effective ”