MANTRA: The Manifold Triangulations Assemblage

TL;DR

MANTRA introduces a large-scale, intrinsically higher-order benchmark of triangulations of $2$- and $3$-manifolds to evaluate topology-aware learning methods. The study benchmarks twelve models spanning graph-based and simplicial-complex architectures on tasks such as predicting Betti numbers $\beta_i$, homeomorphism type, and orientability, finding that simplicial models better extract topological invariants but overall performance remains imperfect and sensitive to representation. A central finding is that barycentric subdivisions degrade performance, challenging the notion of remeshing symmetry in current higher-order models, and highlighting the need for new approaches in topological deep learning. MANTRA provides public data and benchmarks to spur the development of robust higher-order methods beyond standard graph treatments, while the absence of vertex coordinates further emphasizes topology-driven evaluation.

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.

Figures (2)

• Figure 1: Geometric realizations of some manifold triangulations included in MANTRA. The precise coordinates of vertices in Euclidean space are not geometrically significant; what matters is the topology of the resulting polyhedra. Hence, MANTRA is a purely combinatorial dataset.
• Figure 2: From left to right, four simplicial complexes $\text{K}_1$, $\text{K}_2$, $\text{K}_3$, and $\text{K}_4$ with their respective Betti numbers $\beta_0$, $\beta_1$, and $\beta_2$. The $n$-th Betti number indicates the number of $n$-dimensional holes in a geometric realization of a simplicial complex. Here $\text{K}_1$ is a solid tetrahedron with $\beta_0=1$, $\beta_1=0$, and $\beta_2=0$, since $\text{K}_1$ has only one connected component, no unfilled cycles, and no empty cavity enclosed by $2$-faces; $\text{K}_2$ is a hollow tetrahedron with $\beta_0=1$, $\beta_1=0$, and $\beta_2=1$ (the difference with $\text{K}_1$ is that the triangles of $\text{K}_2$ enclose a cavity); $\text{K}_3$ is the underlying graph, with $\beta_0=1$, $\beta_1=3$, and $\beta_2=0$, since there is no cavity and there are three linearly independent cycles; $\text{K}_4$ consists of four vertices and has $\beta_0=4$, $\beta_1=0$, and $\beta_2=0$, since there are four connected components and no cycles nor cavities.