funFMC: Overlapping Clustering for Functional Data

2026-07-14

AI Summary

This study addresses the challenge that conventional exclusive clustering methods fail to capture the prevalent overlapping cluster structures in multivariate functional data. To overcome this limitation, the authors propose a novel approach based on a latent factor model, which jointly models cluster memberships and their cardinalities through functional factors and a real-valued loading matrix. This work establishes, for the first time in functional clustering, an identifiable framework for overlapping structures by proving the identifiability of the loading matrix up to permutation and deriving a corresponding central limit theorem to support statistical inference. By leveraging the inner product structure of Hilbert–Schmidt operator spaces, the infinite-dimensional operator regression is transformed into a finite-dimensional real matrix optimization problem. Numerical experiments and analyses of fMRI data demonstrate that the proposed method accurately recovers overlapping clusters and enables valid statistical inference.

Abstract

In applications such as neuroscience and environmental science, data are naturally modeled as multivariate functional data and often exhibit overlapping cluster structure. Existing clustering methods for functional data typically impose mutually exclusive memberships and therefore fail to capture such structure. We propose a latent factor model based approach with functional factors and a real-valued loading matrix that encodes potentially overlapping cluster memberships. Under mild conditions, we establish identifiability of the loading matrix up to permutation, ensuring that the overlapping cluster structure is recoverable up to label switching. We develop a procedure for estimating both the number of clusters and the associated cluster memberships. This involves solving an infinite-dimensional regression problem in operators, whose solution is characterized using the inner product on the space of Hilbert-Schmidt operators and expressed in terms of real-valued matrices. This formulation enables rigorous asymptotic analysis, and we establish a central limit theorem to facilitate statistical inference on overlapping cluster memberships. We demonstrate the performance of our method using numerical studies and an application to functional magnetic resonance imaging data.

Problem

• functional data
• overlapping clustering
• cluster structure
• multivariate functional data
• cluster membership

Innovation

• overlapping clustering
• functional data
• latent factor model
• Hilbert-Schmidt operators
• identifiability