  Connectomics has emerged as a powerful tool in neuroimaging and has spurred
recent advancements in statistical and machine learning methods for
connectivity data. Despite connectomes inhabiting a matrix manifold, most
analytical frameworks ignore the underlying data geometry. This is largely
because simple operations, such as mean estimation, do not have easily
computable closed-form solutions. We propose a geometrically aware neural
framework for connectomes, i.e., the mSPD-NN, designed to estimate the geodesic
mean of a collections of symmetric positive definite (SPD) matrices. The
mSPD-NN is comprised of bilinear fully connected layers with tied weights and
utilizes a novel loss function to optimize the matrix-normal equation arising
from Fr\'echet mean estimation. Via experiments on synthetic data, we
demonstrate the efficacy of our mSPD-NN against common alternatives for SPD
mean estimation, providing competitive performance in terms of scalability and
robustness to noise. We illustrate the real-world flexibility of the mSPD-NN in
multiple experiments on rs-fMRI data and demonstrate that it uncovers stable
biomarkers associated with subtle network differences among patients with
ADHD-ASD comorbidities and healthy controls.
