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

Abstract

This paper extends the blurring mean shift algorithm from vector-valued data to functional data, enabling effective clustering in infinite-dimensional settings without requiring specification of the number of clusters. 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 convergence analysis for the full blurring functional mean shift procedure, establishing theoretical guarantees for its iterative behavior. For the stochastic variant, we provide partial theoretical justification by showing that, when the subset size is sufficiently large, its one-step update is well approximated by the corresponding update of the full algorithm. The proposed method is demonstrated through real-data applications, including hourly Taiwan PM$_{2.5}$ measurements and Argo oceanographic profiles. Our key contributions include: (1) extending the blurring mean shift algorithm to functional data in a Hilbert-space setting; (2) developing a scalable stochastic variant based on random partitioning for large-scale data; (3) establishing convergence results for the full blurring functional mean shift algorithm; and (4) demonstrating the scalability and practical usefulness of the proposed method through simulation and real-data applications.