论文标题
几何学和自适应$ k $ - 均值渐近理论
Asymptotic Theory of Geometric and Adaptive $k$-Means Clustering
论文作者
论文摘要
我们重新审视Pollard在欧几里得空间中聚类的$ k $ - 均值一致性的经典结果，重点关注两个方向的扩展：首先，数据可能来自有趣的几何几何环境（例如，Riemannian歧管，反射性Banach空间，或Wasserstein Space）；其次，从数据（例如$ k $ -Medoids或erbow-method $ k $ -means）中选择一些参数的问题。为此，我们提供了一种一般理论，该理论表明上述所有聚类程序都是强烈一致的。实际上，我们的证明方法使我们能够得出许多超出强度一致性的渐近极限定理。我们还删除了有关最佳集群中心集合的唯一性的所有假设。
We revisit Pollard's classical result on consistency for $k$-means clustering in Euclidean space, with a focus on extensions in two directions: first, to problems where the data may come from interesting geometric settings (e.g., Riemannian manifolds, reflexive Banach spaces, or the Wasserstein space); second, to problems where some parameters are chosen adaptively from the data (e.g., $k$-medoids or elbow-method $k$-means). Towards this end, we provide a general theory which shows that all clustering procedures described above are strongly consistent. In fact, our method of proof allows us to derive many asymptotic limit theorems beyond strong consistency. We also remove all assumptions about uniqueness of the set of optimal cluster centers.
