TopologyUnderstandingNet (TUN): Geometry-Conditioned Classification of Persistence Diagram Points

Persistence diagrams (PDs) summarize topological changes across a filtration, but deciding which individual entries reflect source structure rather than sampling artifacts remains difficult because persistence alone does not encode the geometry that generated the diagram. We formulate this problem as point-wise significance classification and construct a supervised benchmark spanning planar shapes, CAD models, triply periodic minimal surfaces, and zeolite frameworks. Its labels combine known topological counts with instance-specific geometric review, supported by a conditional analysis of an early-birth criterion. We then propose TopologyUnderstandingNet (TUN), which models interactions among diagram entries and conditionally refines each prediction using a point-cloud representation of the source geometry. Experiments show that TUN achieves the highest average classification performance and improves over its matched PD-only counterpart across all four categories. In aggregate CAD source-identity retrieval, equal-size subdiagrams selected by TUN also outperform persistence ranking, indicating that geometry-conditioned predictions preserve source information not captured by persistence alone.