BIRS Workshop Lecture Videos
Random Matrices with Heavy-Tailed Entries: Tight Mean Estimators and Applications to Statistics Minsker, Stanislav
Estimation of the covariance matrix has attracted significant 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, or ``atypical’’ points in the sample. As P. 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 Tukey's 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. Our arguments rely on generic chaining techniques applied to operator-valued stochastic processes, as well as bounds on the trace moment-generating function. We will discuss extensions of our approach to matrix-valued U-statistics and examples such as matrix completion problem. Part of the talk will be based on a joint work with Xiaohan Wei.
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