Estimation of the covariance structure of heavy-tailed distributions
Estimation of the covariance structure of heavy-tailed distributions
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发表时间：
2017-08
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通讯作者：
Xiaohan Wei;Stanislav Minsker
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作者：
Xiaohan Wei;Stanislav Minsker
我们提出和分析了协方差矩阵的新估计器，该矩阵在对基本分布的假设较弱的情况下（例如仅存在低阶的矩存在），该协方差矩阵承认了强大的理论保证。尽管对相对于高斯分布的协方差矩阵的估计是充分理解的，但在重尾数据的情况下却少得多。正如K. Balasubramanian和M. Yuan所写的那样，“现实世界实验的数据常常往往会被异常值和/或表现出沉重的尾巴损坏。在这种情况下，尚不清楚这些协方差矩阵估计器。 “ ..处理重型分布的其他可能策略是什么，需要进一步研究。”我们朝着回答这个问题迈出了一步，并证明了仅取决于控制与协方差矩阵相关的“内在维度”的参数（与环境空间的维度相反）的参数紧密偏差不等式；特别是，我们的结果适用于高维观测。
We propose and analyze a new estimator of the covariance matrix that admits strong theoretical guarantees under weak assumptions on the underlying distribution, such as existence of moments of only low order. While estimation of covariance matrices corresponding to sub-Gaussian distributions is well-understood, much less in known in the case of heavy-tailed data. As K. Balasubramanian and M. Yuan write, "data from real-world experiments oftentimes tend to be corrupted with outliers and/or exhibit heavy tails. In such cases, it is not clear that those covariance matrix estimators .. remain optimal" and "..what are the other possible strategies to deal with heavy tailed distributions warrant further studies." We make a step towards answering this question and prove tight deviation inequalities for the proposed estimator that depend only on the parameters controlling the "intrinsic dimension" associated to the covariance matrix (as opposed to the dimension of the ambient space); in particular, our results are applicable in the case of high-dimensional observations.