No Triangulation Without Representation: Generalization in Topological Deep Learning

2026-05-07

AI Summary

This work addresses the lack of a unified evaluation framework for assessing whether existing topological deep learning models genuinely capture higher-order topological structures. To this end, the authors extend the MANTRA benchmark to encompass a broader range of manifold types and introduce a novel evaluation protocol based on representation diversity and mesh refinement. Using this framework, they systematically evaluate the generalization capabilities of graph neural networks (GNNs) and higher-order message passing (HOMP) approaches. Their experiments reveal that while carefully designed representations can achieve near-saturated performance on current benchmarks, these models fail to generalize across topological structures at different scales. This limitation underscores the absence of scale-invariant topological inductive biases in existing methods and highlights a critical direction for future research.

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.

Problem

• topological deep learning
• generalization
• triangulation
• representation
• manifold

Innovation

• topological deep learning
• manifold triangulation
• representational diversity
• generalization
• evaluation benchmark