Sub-Gaussian estimators of the mean of a random matrix with heavy-tailed entries

By Stanislav Minsker. Annals of Statistics, 2016

115 citations recorded by the source corpus.

Estimation of the covariance matrix has attracted a lot of attention of the statistical research community over the years, partially due to important applications such as Principal Component Analysis. However, frequently used empirical covariance estimator (and its modifications) is very sensitive to outliers in the data. As P. J. Huber wrote in 1964, "...This raises a question which could have been asked already by Gauss, but which was, as far as I know, only raised a few years ago (notably by Tukey): what happens if the true distribution deviates slightly from the assumed normal one? As is now well known, the sample mean then may have a catastrophically bad performance..." Motivated by this question, we develop a new estimator of the (element-wise) mean of a random matrix, which includes covariance estimation problem as a special case. Assuming that the entries of a matrix possess only finite second moment, this new estimator admits sub-Gaussian or sub-exponential concentration around the unknown mean in the operator norm. We will explain the key ideas behind our construction, as well as applications to covariance estimation and matrix completion problems.

Subfields: Computational Mechanics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e5d38bf1b7726ee42b3c83ef85a8421239460362

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In Computational Mechanics

In Statistics and Probability

Citations in the corpus

Distributed Statistical Estimation and Rates of Convergence in Normal Approximation (2017, background): “Recently, Lugosi and Mendelson (2017, 2018) proposed new estimators that attains optimal bounds and requires existence of only 2 moments.”
Robust covariance estimation under $L_{4}-L_{2}$ norm equivalence (2018, result): “1 there and similar results in [10, 11]), in which X is assumed to satisfy an L4 − L2 norm equivalence.”