  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.
