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📄 Abstract
Abstract: We present two sharp, closed-form empirical Bernstein inequalities for
symmetric random matrices with bounded eigenvalues. By sharp, we mean that both
inequalities adapt to the unknown variance in a tight manner: the deviation
captured by the first-order $1/\sqrt{n}$ term asymptotically matches the matrix
Bernstein inequality exactly, including constants, the latter requiring
knowledge of the variance. Our first inequality holds for the sample mean of
independent matrices, and our second inequality holds for a mean estimator
under martingale dependence at stopping times.