Horizontal Flows and Manifold Stochastics in Geometric Deep Learning.

Area of Science:

• Geometric Deep Learning
• Stochastic Differential Geometry
• Manifold Learning

Background:

• Convolutional Neural Networks (CNNs) excel on Euclidean data but struggle with non-Euclidean manifolds.
• Existing geometric deep learning methods often lack continuous filter transport and efficient evaluation strategies.
• Holonomy effects, crucial for orientation-dependent data on manifolds, are challenging to incorporate.

Purpose of the Study:

• To introduce two novel constructions for geometric deep learning on manifolds.
• To enable continuous transport of orientation-dependent convolutional filters, naturally incorporating holonomy.
• To facilitate efficient evaluation of manifold convolution layers via stochastic sampling.

Main Methods:

• Developing a continuous filter transport mechanism using horizontal frame bundle flows.
• Implementing efficient layer evaluation through sampling manifold-valued random variables around a weighted diffusion mean.
• Leveraging stochastic methods inspired by geometric statistics and manifold stochastic processes.

Main Results:

• A novel convolution operator on manifolds that inherently accounts for holonomy.
• An efficient sampling scheme for manifold convolution layers based on non-linear bridge sampling.
• Theoretical foundations and properties of the proposed geometric deep learning constructions.

Conclusions:

• Stochastic methods offer powerful tools for advancing geometric deep learning.
• The proposed constructions provide a robust framework for deep learning on manifolds.
• These methods enhance the capability of deep learning models to handle complex, orientation-dependent data structures.