  We consider the problem of constructing nonparametric undirected graphical
models for high-dimensional functional data. Most existing statistical methods
in this context assume either a Gaussian distribution on the vertices or linear
conditional means. In this article we provide a more flexible model which
relaxes the linearity assumption by replacing it by an arbitrary additive form.
The use of functional principal components offers an estimation strategy that
uses a group lasso penalty to estimate the relevant edges of the graph. We
establish statistical guarantees for the resulting estimators, which can be
used to prove consistency if the dimension and the number of functional
principal components diverge to infinity with the sample size. We also
investigate the empirical performance of our method through simulation studies
and a real data application.
