[Paper Review] Combinatorial Persistent Homology Transform

Brittany Terese Fasy, Amit Patel|arXiv (Cornell University)|Aug 10, 2022

Topological and Geometric Data Analysis18 references4 citations

TL;DR

This paper introduces a combinatorial reformulation of the Persistent Homology Transform (PHT) as a cellular cosheaf of persistence diagrams, leveraging functorial persistent homology over abstract posets. By indexing filtrations on a finite cellulation of the sphere $\mathbb{S}^N$, the PHT becomes a finite, combinatorially structured cosheaf, enabling categorical and sheaf-theoretic tools for shape analysis with exact, stable representations of geometric complexes.

ABSTRACT

The combinatorial interpretation of the persistence diagram as a Möbius inversion was recently shown to be functorial. We employ this discovery to recast the Persistent Homology Transform of a geometric complex as a representation of a cellulation on $\mathbb{S}^n$ to the category of combinatorial persistence diagrams. Detailed examples are provided. We hope this recasting of the PH transform will allow for the adoption of existing methods from algebraic and topological combinatorics to the study of shapes.

Motivation & Objective

• To address the limitation of traditional PHTs, which produce uncountably many distinct persistence diagrams even for finitely generated complexes.
• To overcome the categorical mismatch between combinatorially equivalent but topologically distinct diagrams in real-indexed persistent homology.
• To develop a finite, cellular representation of the PHT using combinatorial persistent homology over abstract posets.
• To recast the PHT as a cellular cosheaf, enabling application of sheaf theory and category theory to shape analysis.
• To provide a foundation for future study of cosheaf homology, global sections, and stability in parameterized persistence.

Proposed method

• Adopt the combinatorial persistent homology framework of McClearly and Patel, indexing filtrations on an abstract totally ordered poset instead of $\mathbb{R}$.
• Construct a finite cellulation of $\mathbb{S}^N$ such that directions within the same cell yield identical combinatorial persistence diagrams.
• Define the PH transform as a cosheaf $F: \mathcal{C} \to \mathsf{Fnc}$, where $\mathcal{C}$ is the cellulation and $\mathsf{Fnc}$ is the category of persistence modules.
• Use face relations in the cellulation to induce poset maps between vertex equivalence classes, which generate the cosheaf structure.
• Leverage functoriality of persistent homology to ensure compatibility of diagrams across cell inclusions.
• Apply the theory of cellular cosheaves to represent the entire PHT as a finite, cohesive object amenable to algebraic and topological analysis.

Experimental results

Research questions

• RQ1 Can the Persistent Homology Transform be recast as a cellular cosheaf of combinatorial persistence diagrams to enable categorical analysis of shapes?
• RQ2 How does the finite cellulation of $\mathbb{S}^N$ ensure that directions in the same cell yield combinatorially equivalent persistence diagrams?
• RQ3 What is the role of functoriality in constructing a well-defined cosheaf structure from the PHT?
• RQ4 How can existing tools from (co)sheaf theory and category theory be applied to the resulting cosheaf of persistence diagrams?
• RQ5 What are the global sections and cohomological invariants of such a cosheaf, and how do they relate to shape reconstruction?

Key findings

• The PH transform of a PL-embedded finite simplicial complex admits a finite cellular decomposition of $\mathbb{S}^N$ such that all directions in a single cell yield identical combinatorial persistence diagrams.
• The PHT is isomorphic to a cellular cosheaf of combinatorial persistence diagrams, making it a finite object in the category of cosheaves.
• The cosheaf structure arises from face relations in the cellulation, which induce poset maps between vertex equivalence classes under height functions.
• Type-1 and type-2 zero-cells correspond to three-vertex and two-pair equivalence classes, respectively, with non-existent labelings arising from topological contradictions.
• The cosheaf framework allows for a unified, cohesive representation of all persistence diagrams across directions, enabling global analysis via category theory.
• The approach provides a stable, complete, and finite representation of the original shape, resolving the uncountability and non-isomorphism issues of traditional PHTs.