Horizontal Flows and Manifold Stochastics in Geometric Deep Learning
This work addresses challenges in geometric deep learning for processing data on curved surfaces, representing an incremental advancement by building on existing stochastic methods.
The paper tackles the problem of defining convolution operators on manifolds that incorporate rotational effects of holonomy and enabling efficient evaluation through sampling manifold-valued random variables, resulting in methods inspired by stochastics on manifolds and geometric statistics.
We introduce two constructions in geometric deep learning for 1) transporting orientation-dependent convolutional filters over a manifold in a continuous way and thereby defining a convolution operator that naturally incorporates the rotational effect of holonomy; and 2) allowing efficient evaluation of manifold convolution layers by sampling manifold valued random variables that center around a weighted diffusion mean. Both methods are inspired by stochastics on manifolds and geometric statistics, and provide examples of how stochastic methods -- here horizontal frame bundle flows and non-linear bridge sampling schemes, can be used in geometric deep learning. We outline the theoretical foundation of the two methods, discuss their relation to Euclidean deep networks and existing methodology in geometric deep learning, and establish important properties of the proposed constructions.