We present two sharp 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.

翻译：我们针对特征值有界的对称随机矩阵提出了两个尖锐的经验Bernstein不等式。所谓"尖锐"，是指两个不等式均能以紧致方式适应未知方差：一阶$1/\sqrt{n}$项所捕获的偏差在渐近意义上精确匹配矩阵Bernstein不等式（包括常数项），而后者需要已知方差信息。我们的第一个不等式适用于独立矩阵的样本均值，第二个不等式适用于停时处鞅依赖条件下的均值估计量。