Abstract
Developing deep neural networks (DNNs) for manifold-valued data sets has gained significant interest of late in the deep learning research community. Examples of manifold-valued data in the medical imaging domain include (but are not limited to) diffusion magnetic resonance imaging, tensor-based morphometry, shape analysis and more. In this paper we present a novel theoretical framework for DNNs to cope with manifold-valued data inputs, taking inspiration from the convolutional neural network (CNN) architecture. We call our network the ManifoldNet.
Analogous to vector spaces where convolutions are equivalent to computing weighted means, manifold-valued data convolutions can be defined using the weighted Fréchet Mean (wFM). To this end, we present a provably convergent recursive algorithm for computation of the wFM of the given data, where the weights are to be learned. Further, we prove that the proposed wFM layer achieves a contraction mapping and hence the ManifoldNet need not have additional non-linear ReLU units used in standard CNNs to achieve a contraction mapping.
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 regression between diffusion MRI scans of Parkinson Disease (PD) patients and clinical information such as their Movement Disorder Society’s Unified Parkinson’s Disease Rating Scale (MDS-UPDRS) scores. In another experiment, we present results of finding group differences based on brain connectivity at the fiber bundle level between PD and controls.
This research was in part funded by the NSF grants IIS-1525431 and IIS-1724174.
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Chakraborty, R., Bouza, J., Manton, J., Vemuri, B.C. (2019). A Deep Neural Network for Manifold-Valued Data with Applications to Neuroimaging. In: Chung, A., Gee, J., Yushkevich, P., Bao, S. (eds) Information Processing in Medical Imaging. IPMI 2019. Lecture Notes in Computer Science(), vol 11492. Springer, Cham. https://doi.org/10.1007/978-3-030-20351-1_9
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