均值和协方差的不可知估计
Agnostic Estimation of Mean and Covariance
摘要：
我们考虑的问题估计的平均值和协方差分布从IID样本$\mathbb{R}^n$，在存在的恶意噪声的$η$分数，这是在最近的工作，噪声本身被假定为从一个已知类型的分布。不可知论问题包括许多有趣的特殊情况，例如，当$η$部分数据被逆向损坏时，学习单个高斯（或找到最佳拟合高斯）的参数，不可知地学习高斯的混合物，不可知ICA等。我们提出了多项式时间算法来估计均值和协方差，并在信息理论下限方面具有误差保证。作为推论，我们还得到了奇异值分解的一个不可知算法。
英文摘要：
We consider the problem of estimating the mean and covariance of a distribution from iid samples in $\mathbb{R}^n$, in the presence of an $η$ fraction of malicious noise; this is in contrast to much recent work where the noise itself is assumed to be from a distribution of known type. The agnostic problem includes many interesting special cases, e.g., learning the parameters of a single Gaussian (or finding the best-fit Gaussian) when $η$ fraction of data is adversarially corrupted, agnostically learning a mixture of Gaussians, agnostic ICA, etc. We present polynomial-time algorithms to estimate the mean and covariance with error guarantees in terms of information-theoretic lower bounds. As a corollary, we also obtain an agnostic algorithm for Singular Value Decomposition.
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