Look Before You Lift: Visual and Quantitative Diagnostics for Topological Deep Learning

Mathilde Papillon; Guillermo Bernárdez; Álvaro Ballón Barreiro; Marco Montagna; Rémi Devaux; Antoine Jardin; Nina Miolane

cs.LGAug 15, 2026

Look Before You Lift: Visual and Quantitative Diagnostics for Topological Deep Learning

Authors: Mathilde Papillon, Guillermo Bernárdez, Álvaro Ballón Barreiro, Marco Montagna, Rémi Devaux, Antoine Jardin, Nina Miolane

Organizations: University of California Santa Barbara · Sapienza Universitá di Roma · Arlequin AI

Abstract

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 ${\mathcal{T}}$ (left), three examples of strictly augmented Hasse graphs $\mathcal{G}_{\mathcal{N}}$ corresponding to 4 neighborhood functions: adjacency of nodes w.r.t. edges ( ${\mathcal{A}}_{0,1}$ ), incidence from nodes to edges ( ${\mathcal{I}}_{0\to 1}$ ), incidence from faces to nodes ( ${\mathcal{I}}_{2\to 1}$ ), and incidence from nodes to faces ( ${\mathcal{I}}_{0\to 2}$ ). 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 $\rho$ between Hasse graph metrics and normalized model performance. We explore (left) spectral radius $\lambda_{\max}$ on adjacency Hasse graphs $\mathcal{G}_{\mathcal{A}_{t,s}}$ , (middle) spectral radius on incidence Hasse graphs $\mathcal{G}_{\mathcal{I}_{s\to t}}$ , 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 $\lambda_{\max}(\mathcal{G}_{\mathcal{N}})$ and clustering coefficient $C(\mathcal{G}_{\mathcal{N}})$ 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 $m/\binom{n}{2}$ for $n>1$ ; $0$ if $n\leq 1$ .
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.

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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

\partial

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

\partial

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.

Abstract

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