Semantic grounding and interpretability of topological features

Establish methods for semantic grounding that connect persistent homology features—such as cycles, cavities, and other homology classes—to stable, domain-legible structures in the underlying system, resolving the non-uniqueness of representatives and enabling interpretable, mechanism-relevant explanations.

Background

The authors note that persistence diagrams summarize existence and lifetimes of homological features but usually do not identify unique, semantically interpretable representatives in data.

They highlight non-uniqueness of cycle representatives and recent attempts (e.g., stable volumes) to link features back to concrete structures, arguing that these efforts are not yet a general solution.

They frame the key need as mapping topological summaries to stable, domain-meaningful objects that clarify what reorganized in complex systems.

References

The real open problem is semantic grounding: connecting a topological feature to a stable, domain-legible structure in the underlying system.

— Topology as a Language for Emergent Organization in Complex Systems: Multiscale Structure, Higher-Order Interactions, and Early Warning Signals  (2603.25760 - Bailey, 25 Mar 2026) in Section 8.3: Interpretability remains a genuine bottleneck

Conversely, the Province of Bari exhibits a more diffuse topological organization associated with a larger diversity of competing persistence scales. The proposed interpretation remains to be validated through additional analyses and will be explored in future developments of this study.

— Topological Uncertainty and Higher-Order Interactions in Spatial Networks  (2609.18439 - Pomarico et al., 16 Sep 2026) in Discussion, paragraph beginning “The persistence results reveal substantial differences between the Bari and Gargano case studies.”