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

1. Introduction

• Here, TDL refers to methods that learn on discrete higher-order domains; methods using tools such as persistent homology are outside the paper’s scope.Source: §1 · p. 1
• Different liftings can induce drastically different connectivity on the same dataset, motivating inspection of lifted structure before training.Strictly augmented Hasse graph decompositions expose incidence- and adjacency-based neighborhoods across cell ranks.Source: §1 · p. 2
• TopoExplorer is a visual and quantitative diagnostic framework that extracts structural and feature-based graph metrics from lifted complexes.It reframes the workflow from lift-train to lift-look-design-train.Source: §1 · p. 2
• Experiments using TopoBench’s default liftings show that some pre-training metrics computed in TopoExplorer correlate with downstream model performance.The paper presents this as evidence for a structure-aware design pipeline.Source: §1 · p. 2

• A combinatorial complex is (V, C, rk), with C a subset of nonempty subsets of V; every singleton has rank 0, and rk is order-preserving under inclusion.Source: §2 · p. 2
• For a neighborhood function N, the strictly augmented Hasse graph contains cells with at least one N-neighbor and directed edges from each neighbor to its cell.Its node set is restricted to cells that have at least one neighbor according to N.Source: §2 · p. 3
• A lifting is a uniform structural map between discrete topological domains that identifies motifs and maps them to target cells, including clique lifting and skeletal projections.Clique lifting maps (k + 1)-cliques to k-cells; lifting can also translate between domains of the same dimension or map higher- to lower-order representations.Source: §2 · p. 3
• A neighborhood function maps each cell to neighboring cells; traditional forms are same-rank adjacency via another rank and directed cross-rank incidence.Figure 1 visualizes examples of adjacency and incidence neighborhoods.Source: §2 · p. 3, Figure 1 · p. 3

Higher-Order Representation Learning

• TDL progressed from simplicial and cell-complex networks to combinatorial-complex and modular Hasse-graph architectures, but typically assumes the input complex is fixed (Ebli et al., 2020; Bodnar et al., 2021; Papillon et al., 2025; Bernárdez et al., 2026).TopoTune separates incidence and adjacency neighborhoods for GNN backbones; HOPSE uses the decomposition for linear-cost higher-order positional encodings.Source: §3 · p. 4

The TDL Lifting Ecosystem

• TopoBench standardizes a catalog of more than 30 liftings, but lacks structural diagnostics; existing visualization tools inspect pairwise graphs, not cross-rank neighborhoods (Bernárdez et al., 2024; Telyatnikov et al., 2025).Only one systematic lifting study is reported, and it is specific to hypergraph neural networks.Source: §3 · p. 4

Graph Metrics and Model Performance

• TopoExplorer uses strictly augmented Hasse graphs to compute classical graph metrics and edge-specific Forman–Ricci curvature across higher-order domains.Prior graph-learning research shows that structural metrics can provide actionable signals, including curvature-based approaches addressing over-squashing (Topping et al., 2022; Fesser and Weber, 2024b, 2024a).Source: §3 · pp. 4–5

Topological Metrics and Model Performance

• Hypergraph and simplicial-complex homophily metrics have linked higher-order structure to model performance, but remain domain-specific; TopoExplorer supports broader graph diagnostics across the topological hierarchy (Telyatnikov et al., 2023).It represents arbitrary higher-order domains as strictly augmented Hasse graphs.Source: §3 · p. 5

4. TopoExplorer: Interact and Diagnose Any Lifted Topological Dataset

• TopoExplorer decomposes a lifted complex into strictly augmented Hasse graphs, exposing how information flows through neighborhoods so practitioners can inspect lifting and hyperparameter choices before training.The framework is built on the TopoBench ecosystem and aims to reveal structural implications before computational resources are spent on model training.Source: §4 · p. 5

4.1. Interface Design

• The interface represents incidence relations as directed edges between ranks and adjacency relations as within-rank edges, revealing cross-rank support, orphan cells, potential message-passing sinks, and peer connectivity.Figure 2 illustrates neighborhood-specific relations in a cellular complex using color-coded Hasse-graph edges.Source: §4.1 · p. 5, Figure 2 · p. 6

4.2. Quantitative Signatures

• Connectivity metrics include sparsity and density, degree statistics, and component counts, which can inform channel pruning, flag bottlenecks, and indicate whether structural encodings capture global or localized context.The metrics have no universal thresholds; users should compare candidate liftings, neighborhoods, and hyperparameter configurations, with computed values recordable in an HTML export.Source: §4.2 · p. 6

4.3. Practitioner Workflow

• Practitioners select a dataset and lifting, choose and inspect or combine neighborhood Hasse graphs, then adjust display filters and render or export the visualization.For inductive datasets, users can navigate samples and the lifting is reapplied automatically; the open-source repository can be adapted for custom datasets, liftings, and metrics.Source: §4.3 · p. 7

4.4. TopoExplorer in Action: Structural Diagnostics for Neighborhood Selection

• Higher-order adjacency and incidence neighborhoods show the strongest performance associations; on simplicial datasets, larger spectral radius and clustering generally track stronger gains, while cellular trends are weaker and less consistent.These correlations are suggestive, not evidence that the descriptors reliably improve design choices; further controlled experiments are needed.Source: §4.4 · p. 8, Figure 3 · p. 8
• Selecting a neighborhood renders its Hasse graph alongside spectral radius, clustering coefficient, and other metrics, providing interpretable diagnostics to guide neighborhood selection before training.The case study computes metrics per sample and averages them at dataset level; clustering coefficients are defined only for adjacency neighborhoods because incidence Hasse graphs are bipartite.Source: §4.4 · pp. 8–9

Conclusion

• TopoExplorer makes structural properties of strictly augmented Hasse graphs visible and measurable before training, exposing neighborhood metrics such as spectral radius and clustering coefficient.The framework connects the lifting step to neighborhood-operator selection, which conditions the performance of general TDL architectures.Source: §5 · p. 9
• The case study shows pre-training structural descriptors have domain- and model-dependent associations with downstream performance, providing interpretable signals for the lift-look-design-train workflow.Source: §5 · p. 9

Future Work

• Future work will broaden surfaced descriptors to include geometric and feature-based metrics across neighborhood types and examine their predictive value across more liftings and architectures.Source: §5 · p. 9

Appendix A. Definition of a Neighborhood

• For a t-cell σ, its incidence neighborhood at rank s consists of s-cells containing σ when t < s and s-cells contained in σ when t > s.The definition is given for a combinatorial complex T.Source: Appendix A · p. 14
• An s-cell τ is co-incident of σ if it contains σ and s > t, and incident of σ if it is contained by σ and s < t.Source: Appendix A · p. 14
• The adjacency neighborhood A_{s,t}(σ) contains rank-t cells whose incidence neighborhoods with respect to s intersect; two t-cells are adjacent if both lie in an s-cell.Source: Appendix A · p. 14

Appendix B. TopoExplorer metrics

• Complex- and node-level statistics use the displayed graph after loading, optional lifting, and user-controlled sampling, evaluated as an undirected view.This makes definitions consistent across plain adjacency, incidence/bipartite, and layered neighborhood displays.Source: Appendix B · p. 14
• Neighborhood-level statistics use the full neighborhood matrix for the selected view, before sampling is applied.Examples include rank-k adjacency and cross-rank incidence.Source: Appendix B · p. 14
• Advanced metrics are optional and may be approximated on large graphs, such as betweenness via k-node sampling above ∼800 nodes.Source: Appendix B · p. 14
• Forman-Ricci curvature and edge betweenness appear in edge tooltips and summary tables for extreme edges.Source: Appendix B · p. 14

Appendix C. Full-Page Screenshots of TopoExplorer

• The Metrics tab reports complex- and node-level quantities 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.Source: Table 1 · p. 15
• Figure 5 shows neighborhood selection and representation in TopoExplorer on the ZINC dataset.It is reproduced at full-page scale for legibility.Source: Appendix C · pp. 14, 16, Figure 5 · p. 16
• Figure 6 shows a MANTRA (Ballester et al., 2025) simplicial dataset sample in TopoExplorer, with the metrics panel enlarged.The screenshot is reproduced at full-page scale.Source: Appendix C · p. 14, Figure 6 · p. 17