  Efforts are underway to study ways via which the power of deep neural
networks can be extended to non-standard data types such as structured data
(e.g., graphs) or manifold-valued data (e.g., unit vectors or special
matrices). Often, sizable empirical improvements are possible when the geometry
of such data spaces are incorporated into the design of the model,
architecture, and the algorithms. Motivated by neuroimaging applications, we
study formulations where the data are {\em sequential manifold-valued
measurements}. This case is common in brain imaging, where the samples
correspond to symmetric positive definite matrices or orientation distribution
functions. Instead of a recurrent model which poses computational/technical
issues, and inspired by recent results showing the viability of dilated
convolutional models for sequence prediction, we develop a dilated
convolutional neural network architecture for this task. On the technical side,
we show how the modules needed in our network can be derived while explicitly
taking the Riemannian manifold structure into account. We show how the
operations needed can leverage known results for calculating the weighted
Fr\'{e}chet Mean (wFM). Finally, we present scientific results for group
difference analysis in Alzheimer's disease (AD) where the groups are derived
using AD pathology load: here the model finds several brain fiber bundles that
are related to AD even when the subjects are all still cognitively healthy.

Computer Vision and Pattern Recognition;Machine Learning