🤖 AI Summary
This work addresses the challenge of performing effective geometric deep learning on non-Euclidean data endowed with orbifold structures—quotient spaces that may contain singularities due to symmetry operations. Recognizing that existing methods struggle to handle such topological complexities, the authors propose the first extension of spectral convolution to orbifolds, thereby establishing foundational building blocks for geometric deep learning on this class of spaces. This advancement broadens the range of topological structures amenable to geometric deep learning and offers a novel framework for modeling data exhibiting both symmetries and singularities. The efficacy of the proposed approach is demonstrated through a case study in music theory, where it successfully captures the intrinsic geometry of real-world non-Euclidean data, highlighting its expressive power and practical potential.
📝 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.