对统计学家有用的戴维斯-卡汉定理
A useful variant of the Davis--Kahan theorem for statisticians
摘要：
戴维斯-卡汉定理被用于许多统计过程的分析，以限制人口特征向量和它们的样本版本所跨越的子空间之间的距离。它依赖于一定的相关人口和样本特征值之间的特征值分离条件。我们提出了这个结果的一个变体，它只依赖于人口特征值分离条件，使其更自然和方便的统计环境中的直接应用，并提高在某些情况下的界限。我们还提供了一个扩展的情况下，所研究的矩阵可能是不对称的，甚至是非正方形，其中的兴趣是在相应的奇异向量所跨越的子空间之间的距离。
英文摘要：
The Davis--Kahan theorem is used in the analysis of many statistical procedures to bound the distance between subspaces spanned by population eigenvectors and their sample versions. It relies on an eigenvalue separation condition between certain relevant population and sample eigenvalues. We present a variant of this result that depends only on a population eigenvalue separation condition, making it more natural and convenient for direct application in statistical contexts, and improving the bounds in some cases. We also provide an extension to situations where the matrices under study may be asymmetric or even non-square, and where interest is in the distance between subspaces spanned by corresponding singular vectors.
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