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.
authors: Rudrasis Chakraborty Jose Bouza Jonathan Manton Baba C. Vemuri
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