Spectral Convolution on Orbifolds for Geometric Deep Learning

TL;DR

This paper introduces spectral convolution on orbifolds, expanding geometric deep learning to handle non-Euclidean data with complex topological structures, demonstrated through a music theory example.

Contribution

It presents the novel concept of spectral convolution on orbifolds, enabling deep learning on more complex geometric data structures.

Findings

Spectral convolution on orbifolds is theoretically formulated.

Application to music theory demonstrates practical utility.

Provides a new tool for geometric deep learning on non-Euclidean domains.

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.