ManifoldNet: A Deep Neural Network for Manifold-Valued Data With Applications

Geometric deep learning is a relatively nascent field that has attracted significant attention in the recent past. This is partly due to the ready availability of manifold-valued data. In this paper we present a novel theoretical framework for developing deep neural networks to cope with manifold-valued fields as inputs. We also present a novel architecture to realize this theory and call it a ManifoldNet. Analogous to convolutions in vector spaces which are equivalent to computing weighted sums, manifold-valued data 'convolutions' can be defined using the weighted Frechet Mean (wFM), an intrinsic operation. The hidden layers of ManifoldNet compute wFM of their inputs, where the weights are to be learnt. Since wFM is an intrinsic operation, the processed data remain on the manifold as they propagate through the hidden layers. To reduce the computational burden, we present a provably convergent recursive algorithm for wFM computation. Additionaly, for non-constant curvature manifolds, we prove that each wFM layer is non-collapsible and a contraction mapping. We also prove that the wFM is equivariant to the symmetry group action admitted by the manifold on which the data reside. To showcase the performance of ManifoldNet, we present several experiments from vision and medical imaging.