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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
TL;DR
TDL lifting choices can produce substantially different higher-order connectivity, but practitioners have limited structural visibility before training. TopoExplorer visualizes and quantifies lifted connectivity, and experiments show that some pre-training metrics correlate with downstream model performance.
Problem
TDL lacks a first-class interface for inspecting how alternative liftings shape higher-order connectivity before model training.
Method
TopoExplorer uses strictly augmented Hasse graph decompositions to visualize incidence- and adjacency-based neighborhoods and compute structural and feature-based metrics.
Results
Some pre-training Hasse-graph metrics correlate with downstream model performance, with stronger associations concentrated in higher-order-cell adjacency and incidence neighborhoods.
Takeaways & Limitations
TopoExplorer provides interpretable, domain- and model-dependent signals for comparing lifting and neighborhood choices in the lift-look-design-train workflow.
Takeaways & Limitations
TopoExplorer’s metrics have no universal thresholds and should be interpreted comparatively for a given dataset and candidate configurations.
Abstract
from arXiv · showhide
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.
1. Introduction
TopoExplorer addresses the limited structural inspection in topological deep learning’s lifting workflow by providing visual and quantitative diagnostics for lifted higher-order connectivity. Experiments indicate that some pre-training metrics correlate with downstream model performance, supporting a more structure-aware design pipeline.
- Scope: TDL here concerns methods learning on discrete higher-order domains, excluding topology tools such as persistent homology.
- Motivation: TDL pipelines lift conventional representations into higher-order domains, but lifting choices are often selected with minimal structural inspection.Lifting constructs higher-order elements using connectivity rules or feature-based criteria before model training.
- Motivation: TopoExplorer makes incidence- and adjacency-based neighborhoods explicit in graph form, enabling inspection of connectivity across cell ranks before training.This supports more informed lifting and neighborhood design choices, especially for neighborhood-driven architectures.
- Contributions: TopoExplorer is a visual and quantitative diagnostic framework based on strictly augmented Hasse graph decompositions.It provides an interactive app, open-source code, and structural and feature-based graph metrics for lifted complexes.
- Results: Experiments using TopoBench’s default liftings show that some pre-training metrics computed in TopoExplorer correlate with downstream model performance.The result supports using structure-aware diagnostics to inform lifting design decisions in TDL.
2. Background
This section introduces combinatorial complexes as a general framework for discrete topological domains in TDL, then defines neighborhood functions, strictly augmented Hasse graphs, and lifting transformations. These concepts formalize how higher-order data and its connectivity are represented.
- Combinatorial Complex: A combinatorial complex is a triple (V, C, rk) with cells drawn from the nonempty subsets of V and a rank function rk : C →Z≥0.Every vertex belongs to the complex with rank 0, and rank is order-preserving under set inclusion.
- Strictly augmented Hasse graphs: Each neighborhood function N induces a strictly augmented Hasse graph whose nodes are cells with at least one N-neighbor and whose directed edges encode those neighborhood relations.The graph is defined by CN = {σ ∈C | N(σ)̸ = ∅} and EN = {(τ, σ) | σ, τ ∈CN , τ ∈N(σ)}.
- Lifting: A lifting transformation is a uniformly applied fixed-rule map L : Din →Dout between discrete topological domains.Although often viewed as promoting lower-order data to higher-order complexes, lifting more broadly represents structural translation between topological spaces.
- Neighborhoods: A neighborhood function maps each cell σ ∈C to a collection of neighboring cells N(σ) ⊂C.Adjacencies connect same-rank cells through relations involving another rank, whereas incidences link cells of different source and target ranks.
3. Related Work
Prior work established topological deep learning as higher-order representation learning and increasingly modularized it through strictly augmented Hasse graphs. Related research also catalogued diverse lifting strategies and structural diagnostics, while leaving a gap in visualizing cross-rank neighborhoods across arbitrary higher-order domains.
- Higher-Order Representation Learning: TDL generalizes GNNs from pairwise edges to higher-order interactions through simplicial, cell-complex, and combinatorial-complex neural networks.SNNs and CCNNs use the rigid hierarchy of algebraic topology, while combinatorial complexes unify these approaches.
- Higher-Order Representation Learning: Modern TDL architectures use neighborhood operators and strictly augmented Hasse graphs to separate incidence and adjacency channels for GNN backbones.TopoTune introduced this decomposition, and HOPSE uses the same representation.
- The TDL Lifting Ecosystem: No existing visualization tool exposes the cross-rank neighborhood structure of lifted higher-order domains.This limitation distinguishes higher-order lifting inspection from established pairwise network exploration.
- Graph Metrics and Model Performance: Structural metrics provide actionable signals for graph learning, and TopoExplorer extends this diagnostic perspective to higher-order domains using Hasse-graph metrics and edge-specific Forman–Ricci curvature.Related work also links homophily in hypergraphs and simplicial complexes to higher-order neural-network performance, though those metrics remain specialized and domain-specific.
4. TopoExplorer: Interact and Diagnose Any Lifted Topological Dataset
TopoExplorer exposes the structural implications of topological liftings through strictly augmented Hasse graphs, interactive neighborhood views, and quantitative diagnostics before model training. Its comparative metrics provide interpretable signals for neighborhood selection and architecture design, while experiments find associations between higher-order connectivity descriptors and downstream performance.
- Framework: TopoExplorer decomposes a lifted complex into strictly augmented Hasse graphs, exposing incidence- and adjacency-based information flow before computational resources are spent on training.The framework is built on the TopoBench ecosystem and supports interactive visual and quantitative diagnosis.
- Quantitative diagnostics: TopoExplorer reports connectivity metrics including edge sparsity and density, degree statistics, and connected-component counts to inform pruning, normalization, attention, and structural-encoding choices.Highly skewed degree statistics flag topological bottlenecks, while connected components indicate whether structural encodings capture global or localized context.
- Quantitative diagnostics: These metrics have no universal thresholds and should be compared across candidate liftings, neighborhood choices, and hyperparameter configurations for the same dataset.An HTML export records computed metrics for comparison across configurations.
- Neighborhood inspection: Users can inspect graph adjacency, graph incidence, higher-order adjacency, and higher-order incidence neighborhoods individually or in combination.Each neighborhood is represented as a Hasse graph, enabling comparison of information flow across ranks.
- Pre-training utility: Experiments associate higher-order-cell neighborhoods, especially 2-cell adjacencies and incidence neighborhoods, with downstream model performance, while noting that controlled experiments are needed to establish design utility.The case study evaluates spectral radius λmax(GN ) and clustering coefficient C(GN ) against TopoTune and HOPSE performance across 20 benchmark datasets.
- Pre-training utility: Selecting a neighborhood automatically renders its Hasse graph and structural metrics, providing interpretable diagnostics that can guide neighborhood selection before training.Available descriptors include spectral radius λmax(GN ) and clustering coefficient C(GN ); clustering coefficients are defined only for adjacency neighborhoods because incidence Hasse graphs are bipartite.
5. Conclusion and Future Work
TopoExplorer makes strictly augmented Hasse graph structure visible and measurable before model training. Its neighborhood-level metrics connect lifting choices to neighborhood operator selection, while the case study indicates that structural descriptors have domain- and model-dependent associations.
- Diagnostic framework: TopoExplorer is an interactive diagnostic framework for exposing and measuring strictly augmented Hasse graph structural properties before model training.The framework targets the diagnostic gap between data lifting and subsequent model development.
- Diagnostic framework: The interface exposes neighborhood-level metrics, including spectral radius and clustering coefficient.These metrics are presented directly within the application interface.
- Design implications: TopoExplorer bridges lifting with neighborhood operator selection, which conditions the performance of general TDL architectures.The connection is made through directly visible neighborhood-level structural metrics.
- Design implications: The case study demonstrates that pre-training structural descriptors carry domain- and model-dependent associations.The supplied passage does not specify the associated metrics or downstream performance values.
Appendix A. Definition of a Neighborhood
The appendix formally defines incidence neighborhoods for cells of a combinatorial complex by rank and containment. It then defines same-rank adjacency through shared incidence relationships, including co-incidence and incidence cases.
- Definition of a Neighborhood: Incidence neighborhoods of a t-cell σ at rank s consist of rank-s cells containing σ when t < s, or contained by σ when t > s.The definition is given with cells τ ∈ C satisfying rk(τ) = s and the corresponding containment relation.
- Definition of a Neighborhood: A rank-s cell τ is co-incident of σ when it contains σ and s > t.This is the higher-rank case of the incidence definition.
- Definition of a Neighborhood: A rank-s cell τ is incident of σ when it is contained by σ and s < t.This is the lower-rank counterpart to co-incidence.
- Definition of a Neighborhood: Same-rank adjacency As,t(σ) comprises cells τ whose incidence neighborhoods overlap, so two t-cells are adjacent when both lie in a common s-cell δ.The adjacency definition is As,t(σ) = {τ ∈ C | rk(τ) = t, Is→t(σ) ∩ Is→t(τ) ≠ ∅}.
Appendix B. TopoExplorer metrics
TopoExplorer computes structural and node-level statistics on the displayed graph after loading, optional lifting, and user-controlled sampling, using an undirected view for consistent definitions across visualization types.
- Displayed-graph statistics: Complex- and node-level statistics are computed on the displayed graph rendered after dataset loading, optional lifting, and user-controlled sampling.The displayed graph is the subgraph shown in the visualization panel.
- Displayed-graph statistics: TopoExplorer reports structural quantities on the graph loaded and configured in its visualization panel.These quantities describe the graph used for visualization and analysis.
- Metric convention: Statistics use an undirected view so the same definitions apply to adjacency, incidence/bipartite, and layered neighborhood displays.This convention standardizes metric definitions across the supported graph representations.
Appendix C. Full-Page Screenshots of TopoExplorer
Appendix C reproduces the two TopoExplorer interface screenshots from the main text at full-page scale, including views of neighborhood selection and metrics on ZINC and MANTRA examples.
- Metric computation: Complex- and node-level quantities are computed on the displayed graph after sampling and lifting, treated as undirected.Neighborhood-level quantities are instead computed on the full neighborhood matrix for each selected view before sampling.
- Full-page screenshots: The appendix provides a full-page version of Figure 2, showing neighborhood selection and representation in TopoExplorer on the ZINC dataset.The screenshot is reproduced at full-page scale for legibility.
- Full-page screenshots: The appendix provides a full-page version of Figure 4, showing TopoExplorer on a MANTRA simplicial dataset sample with the metrics panel enlarged.The figure is one of the two interface screenshots reproduced from the main text.