论文标题
通过经验特征函数几乎最佳的鲁棒平均估计
Nearly Optimal Robust Mean Estimation via Empirical Characteristic Function
论文作者
论文摘要
我们使用经验特征函数提出了一个可分离的实际BANACH空间中随机变量平均值的估计器。假设随机变量的协方差算子在精确的意义上是界定的，我们表明所提出的估计器达到了最佳的亚高斯速率，除了更快的衰减均值依赖性项。在温和的条件下，迭代的完善程序基本上可以消除均值依赖性项并提供换档等级估计值。我们的结果尤其表明，在文献中最著名的速度出现的某种高斯宽度可能是不必要的。此外，使用特征函数的有界性，我们还表明，除了在高信噪比时，提议的估计量可能会优雅地与对抗性“污染”优雅。由于特征功能的障碍，我们的分析是整体简洁而透明的。
We propose an estimator for the mean of random variables in separable real Banach spaces using the empirical characteristic function. Assuming that the covariance operator of the random variable is bounded in a precise sense, we show that the proposed estimator achieves the optimal sub-Gaussian rate, except for a faster decaying mean-dependent term. Under a mild condition, an iterative refinement procedure can essentially eliminate the mean-dependent term and provide a shift-equivariant estimate. Our results particularly suggests that a certain Gaussian width that appears in the best known rate in the literature might not be necessary. Furthermore, using the boundedness of the characteristic functions, we also show that, except possibly at high signal-to-noise ratios, the proposed estimator is gracefully robust against adversarial "contamination". Our analysis is overall concise and transparent, thanks to the tractability of the characteristic functions.
