MANTRA: The Manifold Triangulations Assemblage

2024-10-03

arXiv.org

AI Summary

To address the bottleneck of scarce large-scale, intrinsically high-order benchmark datasets in topological deep learning (TDL), this work introduces the first diverse simplicial complex dataset comprising over 43,000 surfaces and 250,000 triangulations of 3D manifolds. Methodologically, we propose a systematic framework for generating and representing simplicial complexes, and conduct comparative evaluation of graph neural networks (GNNs) and simplicial complex neural networks (SCNNs) on three topological classification tasks. Our key contributions are: (1) the first scalable, intrinsically high-order topological benchmark; (2) empirical evidence demonstrating SCNNs’ superior capability—yet eventual saturation—in capturing fundamental topological invariants; and (3) a critical reexamination of the TDL paradigm, providing a reproducible, verifiable foundation for robust high-order representation learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation
• Social Networks and Social Media: Computational social science

Abstract

The rising interest in leveraging higher-order interactions present in complex systems has led to a surge in more expressive models exploiting higher-order structures in the data, especially in topological deep learning (TDL), which designs neural networks on higher-order domains such as simplicial complexes. However, progress in this field is hindered by the scarcity of datasets for benchmarking these architectures. To address this gap, we introduce MANTRA, the first large-scale, diverse, and intrinsically higher-order dataset for benchmarking higher-order models, comprising over 43,000 and 250,000 triangulations of surfaces and three-dimensional manifolds, respectively. With MANTRA, we assess several graph- and simplicial complex-based models on three topological classification tasks. We demonstrate that while simplicial complex-based neural networks generally outperform their graph-based counterparts in capturing simple topological invariants, they also struggle, suggesting a rethink of TDL. Thus, MANTRA serves as a benchmark for assessing and advancing topological methods, leading the way for more effective higher-order models.

Problem

• Addresses the lack of datasets for benchmarking higher-order models in topological deep learning.
• Introduces MANTRA, a large-scale dataset for evaluating higher-order neural networks.
• Demonstrates the limitations of simplicial complex-based models in capturing topological invariants.

Innovation

• Introduces MANTRA, a large-scale higher-order dataset
• Benchmarks graph- and simplicial complex-based models
• Highlights challenges in topological deep learning