Functional Mixed Membership Models

TL;DR

This paper introduces a Bayesian mixed membership model for functional data, enabling flexible, interpretable clustering of complex signals like EEG data in autism research.

Contribution

It presents a scalable Gaussian process representation using the Karhunen-Loève theorem, with proven posterior consistency and enhanced modeling flexibility.

Findings

Provides a scalable Bayesian framework for functional data analysis.

Formalizes the clinical concept of spectrum in autism EEG studies.

Demonstrates increased interpretability of mean and covariance structures.

Abstract

Mixed membership models, or partial membership models, are a flexible unsupervised learning method that allows each observation to belong to multiple clusters. In this paper, we propose a Bayesian mixed membership model for functional data. By using the multivariate Karhunen-Lo\`eve theorem, we are able to derive a scalable representation of Gaussian processes that maintains data-driven learning of the covariance structure. Within this framework, we establish conditional posterior consistency given a known feature allocation matrix. Compared to previous work on mixed membership models, our proposal allows for increased modeling flexibility, with the benefit of a directly interpretable mean and covariance structure. Our work is motivated by studies in functional brain imaging through electroencephalography (EEG) of children with autism spectrum disorder (ASD). In this context, our work formalizes the clinical notion of "spectrum" in terms of feature membership proportions.