No Triangulation Without Representation: Generalization in Topological Deep Learning

TL;DR

This paper extends a topological dataset benchmark to diverse manifolds, revealing that model performance depends heavily on representation and feature choices, and highlighting the need for models that grasp topological structure beyond data combinatorics.

Contribution

It introduces a new evaluation protocol emphasizing representational diversity, and demonstrates the limitations of current models in generalizing topological structures.

Findings

Both GNNs and HOMP can saturate the benchmark with proper representation.

Model performance depends critically on feature assignment.

Existing models do not generalize beyond combinatorial data structures.

Abstract

Despite an ever-increasing interest in topological deep learning models that target higher-order datasets, there is no consensus on how to evaluate such models. This is exacerbated by the fact that topological objects permit operations, such as structural refinements, that are not appropriate for graph data. In this work, we extend MANTRA, a benchmark dataset containing manifold triangulations, to a larger class of manifolds with more diverse homeomorphism types. We show that, unlike prior claims, both graph neural networks (GNNs) and higher-order message passing (HOMP) methods can saturate the benchmark. However, we find that this is contingent on the right representation and feature assignment, emphasizing their importance in baseline models. We thus provide a novel evaluation protocol based on representational diversity and triangulation refinement. Surprisingly, we find no indication that existing models are capable of generalizing beyond the combinatorial structure of the data. This points towards a research gap in developing models that understand topological structure independent of scale. Our work thus provides the necessary scaffolding to evaluate future models and enable the development of topology-aware inductive biases.