抗扰凸聚类：融合惩罚如何提升抗扰性？
Resistant convex clustering: How does the fusion penalty enhance resistantance?
AI总结：
针对带平方误差损失的凸聚类易受异常值影响的问题，提出抗扰凸聚类方法，理论证明其在超半数观测为异常值时才失效，融合惩罚可增强抗扰性，数值实验验证了其性能优势。
摘要：
凸聚类是k-means与层次聚类的一种凸松弛方法，它通过求解一个凸优化问题实现，目标函数由平方误差损失加上融合惩罚项构成，该惩罚项促使同一簇内观测值的估计质心保持一致。然而，当数据受污染时，采用平方误差损失的凸聚类即使仅存在一个任意异常值也会失效。为应对这一挑战，我们提出了一种抗扰凸聚类方法。理论上，我们证明新估计量对任意异常值具有抗扰性：直到超过半数的观测值为任意异常值时，它才会失效。或许令人意外的是，融合惩罚可通过将估计量融合至未受污染样本的簇中心（而非反向融合）来增强抗扰性。数值研究验证了所提方法的优异性能。
英文摘要：
Convex clustering is a convex relaxation of the $k$-means and hierarchical clustering. It involves solving a convex optimization problem with the objective function being a squared error loss plus a fusion penalty that encourages the estimated centroids for observations in the same cluster to be identical. However, when data are contaminated, convex clustering with a squared error loss fails even when there is only one arbitrary outlier. To address this challenge, we propose a resistant convex clustering method. Theoretically, we show that the new estimator is resistant to arbitrary outliers: it does not break down until more than half of the observations are arbitrary outliers. Perhaps surprisingly, the fusion penalty can help enhance resistance by fusing the estimators to the cluster centers of uncontaminated samples, but not the other way around. Numerical studies demonstrate the competitive performance of the proposed method.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
