  Common measures of brain functional connectivity (FC) including covariance
and correlation matrices are semi-positive definite (SPD) matrices residing on
a cone-shape Riemannian manifold. Despite its remarkable success for
Euclidean-valued data generation, use of standard generative adversarial
networks (GANs) to generate manifold-valued FC data neglects its inherent SPD
structure and hence the inter-relatedness of edges in real FC. We propose a
novel graph-regularized manifold-aware conditional Wasserstein GAN (GR-SPD-GAN)
for FC data generation on the SPD manifold that can preserve the global FC
structure. Specifically, we optimize a generalized Wasserstein distance between
the real and generated SPD data under an adversarial training, conditioned on
the class labels. The resulting generator can synthesize new SPD-valued FC
matrices associated with different classes of brain networks, e.g., brain
disorder or healthy control. Furthermore, we introduce additional population
graph-based regularization terms on both the SPD manifold and its tangent space
to encourage the generator to respect the inter-subject similarity of FC
patterns in the real data. This also helps in avoiding mode collapse and
produces more stable GAN training. Evaluated on resting-state functional
magnetic resonance imaging (fMRI) data of major depressive disorder (MDD),
qualitative and quantitative results show that the proposed GR-SPD-GAN clearly
outperforms several state-of-the-art GANs in generating more realistic
fMRI-based FC samples. When applied to FC data augmentation for MDD
identification, classification models trained on augmented data generated by
our approach achieved the largest margin of improvement in classification
accuracy among the competing GANs over baselines without data augmentation.
