  We consider the problem of estimating the difference between two undirected
functional graphical models with shared structures. In many applications, data
are naturally regarded as a vector of random functions rather than as a vector
of scalars. For example, electroencephalography (EEG) data are treated more
appropriately as functions of time. In such a problem, not only can the number
of functions measured per sample be large, but each function is itself an
infinite-dimensional object, making estimation of model parameters challenging.
This is further complicated by the fact that curves are usually observed only
at discrete time points. We first define a functional differential graph that
captures the differences between two functional graphical models and formally
characterize when the functional differential graph is well defined. We then
propose a method, FuDGE, that directly estimates the functional differential
graph without first estimating each individual graph. This is particularly
beneficial in settings where the individual graphs are dense but the
differential graph is sparse. We show that FuDGE consistently estimates the
functional differential graph even in a high-dimensional setting for both fully
observed and discretely observed function paths. We illustrate the finite
sample properties of our method through simulation studies. We also propose a
competing method, the Joint Functional Graphical Lasso, which generalizes the
Joint Graphical Lasso to the functional setting. Finally, we apply our method
to EEG data to uncover differences in functional brain connectivity between a
group of individuals with alcohol use disorder and a control group.
