Spectral Convolution on Orbifolds for Geometric Deep Learning

Tim Mangliers; Bernhard Mössner; Benjamin Himpel

cs.LGFeb 16, 2026

Spectral Convolution on Orbifolds for Geometric Deep Learning

Authors: Tim Mangliers, Bernhard Mössner, Benjamin Himpel

Organizations: Department of Computer Science Reutlingen University, Germany

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.

Figures & tables

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 $\mathbb{R}^{3}$ .

Panel of Figure 2 — Figure 2: Examples of the real part of some eigenfunctions on $\mathcal{C}^{2}_{12}$ .

Panel of Figure 3 — Figure 3: Visualization of the periodicity function and its symmetrized version.

Panel of Figure 4 — 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.

Geometric Representation Learning

Geometric representation learning encodes data to preserve its intrinsic shape, distances, symmetries, and relationships.

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Modern deep learning architectures increasingly contend with sophisticated signals that are natively infinite-dimensional, such as time series, probability distributions, or operators, and are defined over irregular domains. Yet, a unified learning theory for these settings has been lacking. To start addressing this gap, we introduce a novel convolutional learning framework for possibly infinite-dimensional signals supported on a manifold. Namely, we use the connection Laplacian associated with a Hilbert bundle as a convolutional operator, and we derive filters and neural networks, dubbed as HilbNets. We make HilbNets and, more generally, the convolution operation, implementable via a two-stage sampling procedure. First, we show that sampling the manifold induces a Hilbert Cellular Sheaf, a generalized graph structure with Hilbert feature spaces and edge-wise coupling rules, and we prove that its sheaf Laplacian converges in probability to the underlying connection Laplacian as the sampling density increases. Notably, this result is a generalization to the infinite-dimensional bundle setting of the Belkin & Niyogi \cite{BELKIN20081289} convergence result for the graph Laplacian to the manifold Laplacian, a theoretical cornerstone of geometric learning methods. Second, we discretize the signals and prove that the discretized (implementable) HilbNets converge to the underlying continuous architectures and are transferable across different samplings of the same bundle, providing consistency for learning. Finally, we validate our framework on synthetic and real-world tasks. Overall, our results broaden the scope of geometric learning as a whole by lifting classical Laplacian-based frameworks to settings where the signal at each point lives in its own Hilbert space.

Abstract

Geometric Representation Learning

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