Spectral Convolution on Orbifolds for Geometric Deep Learning
2026-02-16 • Machine Learning
Machine LearningArtificial Intelligence
AI summaryⓘ
The authors explain how to teach computers to learn from complex shapes that aren't simple flat surfaces, like graphs or curved spaces called orbifolds. They introduce a new way to do something called spectral convolution, which is a technique used in deep learning, but now adapted for orbifolds. This helps computers understand data that lives on these complicated shapes. The authors show how this could work with an example from music theory, making it easier to see practical uses.
Geometric Deep LearningSpectral ConvolutionOrbifoldsGraph Neural NetworksManifoldsConvolutional Neural NetworksTopologyGeometryMusic Theory
Authors
Tim Mangliers, Bernhard Mössner, Benjamin Himpel
Abstract
Geometric deep learning (GDL) deals with supervised learning on data domains that go beyond Euclidean structure, such as data with graph or manifold structure. Due to the demand that arises from application-related data, there is a need to identify further topological and geometric structures with which these use cases can be made accessible to machine learning. There are various techniques, such as spectral convolution, that form the basic building blocks for some convolutional neural network-like architectures on non-Euclidean data. In this paper, the concept of spectral convolution on orbifolds is introduced. This provides a building block for making learning on orbifold structured data accessible using GDL. The theory discussed is illustrated using an example from music theory.