  Deep neural networks have become the main work horse for many tasks involving
learning from data in a variety of applications in Science and Engineering.
Traditionally, the input to these networks lie in a vector space and the
operations employed within the network are well defined on vector-spaces. In
the recent past, due to technological advances in sensing, it has become
possible to acquire manifold-valued data sets either directly or indirectly.
Examples include but are not limited to data from omnidirectional cameras on
automobiles, drones etc., synthetic aperture radar imaging, diffusion magnetic
resonance imaging, elastography and conductance imaging in the Medical Imaging
domain and others. Thus, there is need to generalize the deep neural networks
to cope with input data that reside on curved manifolds where vector space
operations are not naturally admissible. In this paper, we present a novel
theoretical framework to generalize the widely popular convolutional neural
networks (CNNs) to high dimensional manifold-valued data inputs. We call these
networks, ManifoldNets.
  In ManifoldNets, convolution operation on data residing on Riemannian
manifolds is achieved via a provably convergent recursive computation of the
weighted Fr\'{e}chet Mean (wFM) of the given data, where the weights makeup the
convolution mask, to be learned. Further, we prove that the proposed wFM layer
achieves a contraction mapping and hence ManifoldNet does not need the
non-linear ReLU unit used in standard CNNs. We present experiments, using the
ManifoldNet framework, to achieve dimensionality reduction by computing the
principal linear subspaces that naturally reside on a Grassmannian. The
experimental results demonstrate the efficacy of ManifoldNets in the context of
classification and reconstruction accuracy.
