Title: MANIFOLDNET: A DEEP NEURAL NETWORK FOR MANIFOLD-VALUED DATA. Abstract: Developing deep neural networks (DNNs) for manifold-valued data sets
has gained much interest of late in the deep learning research
community.  Examples of manifold-valued data include data from
omnidirectional cameras on automobiles, drones etc., diffusion
magnetic resonance imaging, elastography and others. In this paper, we
present a novel theoretical framework for DNNs to cope with
manifold-valued data inputs.  In doing this generalization, we draw
parallels to the widely popular convolutional neural networks (CNNs).
We call our network the ManifoldNet.

As in vector spaces where convolutions are equivalent to computing the
weighted mean of functions, an analogous definition for
manifold-valued data can be constructed involving the computation of
the weighted Fr\'{e}chet Mean (wFM). To this end, we present a
provably convergent recursive computation of the 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 the ManifoldNet does not need the
additional non-linear ReLU unit used in standard CNNs. Operations such
as pooling in traditional CNN are no longer necessary in this setting
since wFM is already a pooling type operation. Analogous to the
equivariance of convolution in Euclidean space to translations, we
prove that the wFM is equivariant to the action of the group of
isometries admitted by the Riemannian manifold on which the data
reside. This equivariance property facilitates weight sharing within
the network.  We present experiments, using the ManifoldNet framework,
to achieve video classification and image reconstruction using an
auto-encoder+decoder setting. Experimental results demonstrate the
efficacy of ManifoldNet in the context of classification and
reconstruction accuracy.