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Spectral Convolution on Orbifolds for Geometric Deep Learning

Tim Mangliers, Bernhard Mössner, Benjamin Himpel

TL;DR

The concept of spectral convolution on orbifolds is introduced and provides a building block for making learning on orbifold structured data accessible using GDL.

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.

Spectral Convolution on Orbifolds for Geometric Deep Learning

TL;DR

The concept of spectral convolution on orbifolds is introduced and provides a building block for making learning on orbifold structured data accessible using GDL.

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.
Paper Structure (12 sections, 1 theorem, 33 equations, 5 figures)

This paper contains 12 sections, 1 theorem, 33 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Geometric Deep Learning
  3. Spectral convolution on orbifolds
  4. Example: Smoothing a periodicity function on the orbifold of dyads
  5. Chord space
  6. Laplacian eigenfunctions on $\mathcal{C}^{2}_{12}$
  7. Smoothing a periodicity function
  8. Discussion and outlook
  9. Discussion
  10. Conceptual implications for geometric deep learning
  11. Perceptual implications of spectral smoothing
  12. Outlook

Key Result

Theorem 3

Let $X$ be a orbifold. Then there exists a unitary operator called the Fourier transform. This operator allows us to define a convolution on $L^2(X)$ by where $\odot$ denotes the pointwise product.

Figures (5)

  • Figure 1: The mapping $\varrho\circ\varphi$ between the fundamental domain $T_1$ of $\mathcal{C}^2_1$ and the Möbius strip as it is classically shown embedded in $\mathbb{R}^{3}$.
  • Figure 2: Examples of the real part of some eigenfunctions on $\mathcal{C}^2_{12}$.
  • Figure 3: Visualization of the periodicity function and its symmetrized version.
  • Figure 4: (a) The symmetrized logarithmic periodicity function on $\mathcal{C}^2_{12}$, (b) the used smoothing filter, and (c) the result of the convolution.
  • Figure 5: Comparison of the symmetrized and smoothed logarithmic periodicity function.

Theorems & Definitions (4)

  • Example 1
  • Example 2
  • Theorem 3
  • Example 4