标题： 函数数据的均值漂移：一种可扩展算法和收敛性分析 显示英文标题

标题： Mean Shift for Functional Data: A Scalable Algorithm and Convergence Analysis

摘要： 本文将均值漂移算法从向量值数据扩展到函数型数据，使得在无限维设置下能够有效聚类。 为应对大规模数据集带来的计算挑战，我们引入了一种快速随机变体，显著降低了计算复杂度。 我们对完整的函数型均值漂移过程进行了严格的收敛性和稳定性分析，建立了其行为的理论保证。 对于随机变体，尽管完整的收敛理论仍待解决，但我们通过展示当子集大小较大时它能很好地近似完整算法，提供了部分使用依据。 所提出的方法通过一个真实的阿戈海洋学剖面数据应用得到了进一步验证。 我们的主要贡献包括：(1) 均值漂移到函数型数据的新扩展；(2) 在希尔伯特空间中完整函数型均值漂移算法的收敛性和稳定性分析；(3) 一种基于随机划分的可扩展随机变体，具有部分理论依据；以及 (4) 一个真实数据应用，展示了该方法的可扩展性和实际有用性。 显示英文摘要

摘要： This paper extends the mean shift algorithm from vector-valued data to functional data, enabling effective clustering in infinite-dimensional settings. To address the computational challenges posed by large-scale datasets, we introduce a fast stochastic variant that significantly reduces computational complexity. We provide a rigorous analysis of convergence and stability for the full functional mean shift procedure, establishing theoretical guarantees for its behavior. For the stochastic variant, although a full convergence theory remains open, we offer partial justification for its use by showing that it approximates the full algorithm well when the subset size is large. The proposed method is further validated through a real-data application to Argo oceanographic profiles. Our key contributions include: (1) a novel extension of mean shift to functional data; (2) convergence and stability analysis of the full functional mean shift algorithm in Hilbert space; (3) a scalable stochastic variant based on random partitioning, with partial theoretical justification; and (4) a real-data application demonstrating the method's scalability and practical usefulness.