  The estimation of functional networks through functional covariance and
graphical models have recently attracted increasing attention in settings with
high dimensional functional data, where the number of functional variables p is
comparable to, and maybe larger than, the number of subjects. In this paper, we
first reframe the functional covariance model estimation as a tuning-free
problem of simultaneously testing p(p-1)/2 hypotheses for cross-covariance
functions. Our procedure begins by constructing a Hilbert-Schmidt-norm-based
test statistic for each pair, and employs normal quantile transformations for
all test statistics, upon which a multiple testing step is proposed. We then
explore the multiple testing procedure under a general error-contamination
framework and establish that our procedure can control false discoveries
asymptotically. Additionally, we demonstrate that our proposed methods for two
concrete examples: the functional covariance model with partial observations
and, importantly, the more challenging functional graphical model, can be
seamlessly integrated into the general error-contamination framework, and, with
verifiable conditions, achieve theoretical guarantees on effective false
discovery control. Finally, we showcase the superiority of our proposals
through extensive simulations and functional connectivity analysis of two
neuroimaging datasets.
