Seminar
Parent Program: | -- |
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Location: | SLMath: Eisenbud Auditorium |
A basic question motivated by applications in statistics and convex geometry is the following: given a mean zero random vector $X$ in $R^n$, how many independent samples $X_1,\ldots X_q$ does it take for the empirical covariance matrix $\tilde{C}=1/q\sum_i X_iX_i^T$ to converge to the actual covariance matrix $\E XX^T$?
In an influential paper, M. Rudelson that if $\|X\|\le O(\sqrt{n})$ a.s. and $\E XX^T=I$, then $O((n\log n)$ samples suffice for an arbitrary fixed constant approximation in the operator norm. Under these very weak assumptions on $X$, this bound is tight.
We show that as long as the k-dimensional marginals of $X$ have bounded 2+\epsilon moments for all k\le n, the logarithmic factor is not needed and O(n) samples are enough, which is the optimal bound.