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.