论文标题
改进的协方差估计：最佳的鲁棒性和次毛司令在沉重的尾巴下保证
Improved covariance estimation: optimal robustness and sub-Gaussian guarantees under heavy tails
论文作者
论文摘要
我们提出了一个从i.i.d.大小$ n $的样本。我们的唯一假设是，该矢量满足了其一维边缘的有限的$ l^p-l^2 $时刻假设，对于某些$ p \ geq 4 $。鉴于此，我们表明可以从样品中估算出$σ$，其高概率错误率与样品协方差矩阵在高斯数据的情况下达到的相同。即使我们允许使用可能没有订单$> p $的时刻的非常一般的发行版，这仍然存在。此外，我们的估计器可以使对抗性污染最佳。这一结果改善了Mendelson和Zhivotovskiy以及Catoni和Giulini的最新贡献，并匹配了Abdalla和Zhivotovskiy的并行作品（在论文中描述了与这项最后一项工作的确切关系）。
We present an estimator of the covariance matrix $Σ$ of random $d$-dimensional vector from an i.i.d. sample of size $n$. Our sole assumption is that this vector satisfies a bounded $L^p-L^2$ moment assumption over its one-dimensional marginals, for some $p\geq 4$. Given this, we show that $Σ$ can be estimated from the sample with the same high-probability error rates that the sample covariance matrix achieves in the case of Gaussian data. This holds even though we allow for very general distributions that may not have moments of order $>p$. Moreover, our estimator can be made to be optimally robust to adversarial contamination. This result improves the recent contributions by Mendelson and Zhivotovskiy and Catoni and Giulini, and matches parallel work by Abdalla and Zhivotovskiy (the exact relationship with this last work is described in the paper).
