Horizontal Flows and Manifold Stochastics in Geometric Deep Learning

Stefan Sommer, Alex Bronstein

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)811-822
Number of pages12
JournalIEEE Transactions on Pattern Analysis and Machine Intelligence
Issue number2
StatePublished - 1 Feb 2022


  • Geometric deep learning
  • bridge sampling
  • curvature
  • frame bundle
  • geometric statistics
  • stochastic analysis on manifolds

All Science Journal Classification (ASJC) codes

  • Software
  • Computer Vision and Pattern Recognition
  • Computational Theory and Mathematics
  • Artificial Intelligence
  • Applied Mathematics


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