Filtration framework for different vertex sets

Develop a framework for comparing filtrations and their Grassmannian persistence diagrams when the filtrations involve different vertex sets rather than a single fixed simplicial complex at which they eventually stabilize.

Background

The paper’s comparison and stability results require two filtrations to be defined on the same finite simplicial complex, with both filtrations eventually stabilizing at that complex. The authors identify extending the framework to filtrations over different vertex sets as an unresolved methodological problem, motivated by the desire to compare a broader class of filtered simplicial data.

References

When comparing two filtrations and their Grassmannian persistence diagrams, we are required that there is a fixed simplicial complex $K$ that each filtration eventually stabilizes at. It is a natural question to ask for a framework that can handle filtrations over different vertex sets.

— Grassmannian Persistence Diagrams: Special Properties in the 1-Parameter Setting  (2504.06077 - Gülen et al., 4 Apr 2025) in Section Discussion