ManifoldNet: A Deep Neural Network for Manifold-valued Data with Applications

Abstract

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 ..