- Series
- Stochastics Seminar
- Time
- Thursday, March 16, 2017 - 3:05pm for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Stas Minsker – University of Southern California
- Organizer
- Christian Houdré
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. We will present extensions of our approach to
matrix-valued U-statistics, as well as applications such as the matrix
completion problem.
Part of the talk will be based on a joint work with Xiaohan Wei.