  Graph theoretical analyses have become standard tools in modeling functional
and anatomical connectivity in the brain. With the advent of connectomics, the
primary graphs or networks of interest are structural connectome (derived from
DTI tractography) and functional connectome (derived from resting-state fMRI).
However, most published connectome studies have focused on either structural or
functional connectome, yet complementary information between them, when
available in the same dataset, can be jointly leveraged to improve our
understanding of the brain. To this end, we propose a function-constrained
structural graph variational autoencoder (FCS-GVAE) capable of incorporating
information from both functional and structural connectome in an unsupervised
fashion. This leads to a joint low-dimensional embedding that establishes a
unified spatial coordinate system for comparing across different subjects. We
evaluate our approach using the publicly available OASIS-3 Alzheimer's disease
(AD) dataset and show that a variational formulation is necessary to optimally
encode functional brain dynamics. Further, the proposed joint embedding
approach can more accurately distinguish different patient sub-populations than
approaches that do not use complementary connectome information.
