# Horizontal Flows and Manifold Stochastics in Geometric Deep Learning

arXiv: https://arxiv.org/abs/1909.06397 · HTML: https://mlsanity.com/paper/1909.06397/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Sep 2019.

## Built on by (most cited first, 6 of 6)

- [Model-centric Data Manifold: the Data Through the Eyes of the Model](https://mlsanity.com/paper/2104.13289/) (2021)
- [A Manifold-based Airfoil Geometric-feature Extraction and Discrepant Data Fusion Learning Method](https://mlsanity.com/paper/2206.12254/) (2022)
- [Deep Learning and Geometric Deep Learning: an introduction for mathematicians and physicists](https://mlsanity.com/paper/2305.05601/) (2023)
- [Manifold GCN: Diffusion-based Convolutional Neural Network for Manifold-valued Graphs](https://mlsanity.com/paper/2401.14381/) (2024)
- [Intrinsic-Metric Physics-Informed Neural Networks (IM-PINN) for Reaction-Diffusion Dynamics on Complex Riemannian Manifolds](https://mlsanity.com/paper/2601.00834/) (2025)
- [Spectral Convolution on Orbifolds for Geometric Deep Learning](https://mlsanity.com/paper/2602.14997/) (2026)

## Abstract (arXiv)

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.