双矩分布的稳健$k$均值聚类
Robust $k$-means Clustering for Distributions with Two Moments
摘要：
我们认为强大的算法的$k$ -均值聚类问题的量化器构造基于$N$独立的意见。我们的主要结果是在一般的可分希尔伯特空间中，基于均值中值的非渐近超额失真界在两个有界矩假设下成立。特别是，我们的研究结果扩展了著名的渐近结果波拉德，1981年谁表明，存在两个时刻是足够的强一致性的经验最佳量化器$\mathbb{R}^d$。在$\mathbb{R}^d$中聚类的特殊情况下，在两个有界时刻下，我们证明了过度失真的匹配（直到常数因子）非渐进上界和下界，这取决于最佳量化器的最轻簇的概率质量。我们的界具有亚高斯形式，证明是基于稳健均值估计的一致界的版本。
英文摘要：
We consider the robust algorithms for the $k$-means clustering problem where a quantizer is constructed based on $N$ independent observations. Our main results are median of means based non-asymptotic excess distortion bounds that hold under the two bounded moments assumption in a general separable Hilbert space. In particular, our results extend the renowned asymptotic result of Pollard, 1981 who showed that the existence of two moments is sufficient for strong consistency of an empirically optimal quantizer in $\mathbb{R}^d$. In a special case of clustering in $\mathbb{R}^d$, under two bounded moments, we prove matching (up to constant factors) non-asymptotic upper and lower bounds on the excess distortion, which depend on the probability mass of the lightest cluster of an optimal quantizer. Our bounds have the sub-Gaussian form, and the proofs are based on the versions of uniform bounds for robust mean estimators.
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