Sep 27, 2018 ICLR 2019 Conference Blind Submission readers: everyone Show Bibtex
  • 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.
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