Topological deep learning (TDL) methods rely on lifting raw data into higher-order discrete domains such as simplicial complexes, cell complexes, and hypergraphs. In practice, this lifting step is often treated as a black box: practitioners select a lifting and then tune architectures, with limited visibility into whether the induced higher-order connectivity is meaningful for the downstream task. To address this missing diagnostic layer, we propose a visualization technique called TopoExplorer that leverages the strictly augmented Hasse graph form of topological datasets for exploratory data analysis. For the first time, practitioners can easily visualize the incidence- and adjacency-based neighborhoods that define the lifted dataset, as well as read off key graph metrics that describe its structural and feature landscape. Via an extensive set of experiments across many datasets and liftings, we show that several of these metrics correlate with downstream model performance, suggesting they can help inform TDL preprocessing design. Our perspective reframes the TDL workflow from lift-train to lift-look-design-train, enabling more principled, interpretable, and efficient model development. TopoExplorer is hosted at https://topoexplorer.pagekite.me, and its source code is available at github.com/geometric-intelligence/topoexplorer.
Figures & tables
Figure 1: Neighborhood expansions of a complex. Given a complex T (left), three examples of strictly augmented Hasse graphs GN corresponding to 4 neighborhood functions: adjacency of nodes w.r.t. edges ( A0,1 ), incidence from nodes to edges ( I0→1 ), incidence from faces to nodes ( I2→1 ), and incidence from nodes to faces ( I0→2 ). Adopted from Bernárdez et al. (2026)
Figure 2: Neighborhood selection and representation in TopoExplorer. Example of a cellular complex from the ZINC ( Irwin et al., 2012 ) dataset, with blue 0-cells (nodes), orange 1-cells (edges) and green 2-cells (faces) represented in terms of Hasse graphs, where the color-coded edges encode neighborhood-specific relations. For a full-page version of this figure, we refer to Appendix C .
Figure 3: Spearman ρ between Hasse graph metrics and normalized model performance. We explore (left) spectral radius λmax on adjacency Hasse graphs GAt,s , (middle) spectral radius on incidence Hasse graphs GIs→t , and (right) clustering coefficient C on adjacency Hasse graphs. The simplicial and cellular domains draw from 12 and 8 datasets, respectively. A ∗ marks significant correlations.
Figure 4: TopoExplorer displaying an interactive rendering of MANTRA ( Ballester et al., 2025 ) simplicial dataset sample (see full-page version in Appendix C ). For each of the 5 selected neighborhoods, the interface (see right panel screenshot) reports the spectral radius λmax(GN) and clustering coefficient C(GN) alongside other metrics (see full list in Appendix B ).
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Metric
Definition
Complex-level metrics (displayed graph)
Nodes
Number of vertices in the displayed graph.
Edges
Number of edges in the displayed graph.
Density
Edge density m/(2n) for n>1 ; 0 if n≤1 .
Avg degree
Mean vertex degree over all nodes.
Max degree
Maximum vertex degree.
Appendix
Table 1: Metrics provided by TopoExplorer’s Metrics tab. Complex- and node-level quantities are computed on the displayed graph (after sampling and lifting), treated as undirected. Neighborhood-level quantities are computed on the full neighborhood matrix for each selected view, before sampling.
Figure 5: Full-page version of Figure 2 : neighborhood selection and representation in TopoExplorer on the ZINC ( Irwin et al., 2012 ) dataset
Figure 6: Full-page version of Figure 4 : TopoExplorer on a MANTRA ( Ballester et al., 2025 ) simplicial dataset sample, with the metrics panel enlarged.
Topological neural networks (TNNs) enable leveraging high-order structures on graphs (e.g., cycles and cliques) to boost the expressive power of message-passing neural networks. In turn, however, these structures are typically identified a priori through an unsupervised graph lifting operation. Notwithstanding, this choice is crucial and may have a drastic impact on a TNN's performance on downstream tasks. To circumvent this issue, we propose ∂lift (DiffLift), a general framework for learning graph liftings to hypergraphs and cellular- and simplicial complexes in an end-to-end fashion. In particular, our approach leverages learned vertex-level latent representations to identify and parameterize distributions over candidate higher-order cells for inclusion. This results in a scalable model which can be readily integrated into any TNN. Our experiments show that ∂lift outperforms existing lifting methods on multiple benchmarks for graph and node classification across different TNN architectures. Notably, our approach leads to gains of up to 45% over static liftings, including both connectivity- and feature-based ones.
Jorge Luiz Franco, Gabriel Duarte, Alexander Nikitin +3
University of São Paulo, Instituto Curvelo · Federal Institute of Ceará · Aalto University +3
After a somewhat rocky start, geometry and topology have established a foothold in machine learning. Message passing, either on graphs or higher-order complexes, is one of the main drivers of geometric deep learning, and paradigms that were once considered to be firmly in the realm of the abstract-like sheaves-have been "tamed" to serve as novel inductive biases for model architectures in topological deep learning. The veritable diversity of models, however, is in stark contrast to the scarcity of suitable benchmark datasets. As a result, researchers often resort to lifting existing graph datasets to include higher-order information. In this opinion paper, I want to encourage the community to also source new datasets, which may be used to prop up the foundations of our research field.
Bastian Rieck
AIDOS Lab, University of Fribourg, Switzerland · Institute of AI for Health, Helmholtz Munich, Germany
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.
Johannes S. Schmidt, Martin Carrasco, Ernst Röell +3
Dept. of Math. & Statistics University of Montréal Montréal, Canada · Dept. of Informatics University of Fribourg Fribourg, Switzerland