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 past few years. This is partly due to the availability of data acquired from non-euclidean domains or features extracted from euclidean-space data that reside on smooth manifolds. For instance, pose data commonly encountered in computer vision reside in Lie groups, while covariance matrices that are ubiquitous in many fields and diffusion tensors encountered in medical imaging domain reside on the manifold of symmetric positive definite matrices. Much of this data is naturally represented as a grid of manifold-valued data. In this paper we present a novel theoretical framework for developing deep neural networks to cope with these grids of manifold-valued data inputs. We also present a novel architecture to realize this theory and call it the ManifoldNet. Analogous to vector spaces where convolutions are equivalent to computing weighted sums, manifold-valued data ‘convolutions’ can be defined using the weighted Fréchet Mean (<inline-formula><tex-math notation="LaTeX">${\sf wFM}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">wFM</mml:mi></mml:math><inline-graphic xlink:href="vemuri-ieq1-3003846.gif"/></alternatives></inline-formula>). (This requires endowing the manifold with a Riemannian structure if it did not already come with one.) The hidden layers of ManifoldNet compute <inline-formula><tex-math notation="LaTeX">${\sf wFM}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">wFM</mml:mi></mml:math><inline-graphic xlink:href="vemuri-ieq2-3003846.gif"/></alternatives></inline-formula>s of their inputs, where the weights are to be learnt. This means the data remain manifold-valued as they propagate through the hidden layers. To reduce computational complexity, we present a provably convergent recursive algorithm for computing the <inline-formula><tex-math notation="LaTeX">${\sf wFM}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">wFM</mml:mi></mml:math><inline-graphic xlink:href="vemuri-ieq3-3003846.gif"/></alternatives></inline-formula>. Further, we prove that on non-constant sectional curvature manifolds, each <inline-formula><tex-math notation="LaTeX">${\sf wFM}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">wFM</mml:mi></mml:math><inline-graphic xlink:href="vemuri-ieq4-3003846.gif"/></alternatives></inline-formula> layer is a contraction mapping and provide constructive evidence for its non-collapsibility when stacked in layers. This captures the two fundamental properties of deep network layers. Analogous to the equivariance of convolution in euclidean space to translations, we prove that the <inline-formula><tex-math notation="LaTeX">${\sf wFM}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">wFM</mml:mi></mml:math><inline-graphic xlink:href="vemuri-ieq5-3003846.gif"/></alternatives></inline-formula> is equivariant to the action of the group of isometries admitted by the Riemannian manifold on which the data reside. To showcase the performance of ManifoldNet, we present several experiments using both computer vision and medical imaging data sets.