[Paper Review] Rank-based persistence

Mattia G. Bergomi, Pietro Vertechi|arXiv (Cornell University)|May 22, 2019

Topological and Geometric Data Analysis4 citations

TL;DR

This paper introduces a unified categorical framework for persistence homology by axiomatizing rank functions in regular and Abelian categories, enabling generalized persistence modules beyond vector spaces or sets. It establishes equality between interleaving and multicolored bottleneck distances in semisimple Abelian categories, enabling structured analysis of data with symmetries or labels—such as group actions or labeled point clouds—via colored persistence diagrams.

ABSTRACT

Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been attempted over time, some topological in flavor, based on the vector space-valued homology functor, other combinatorial, based on arbitrary set-valued functors. To unify the study of topological and combinatorial persistence in a common categorical framework, we give axioms for a generalized rank function on objects in a target category, so that functors to that category induce persistence functions. We port the interleaving and bottleneck distances to this novel framework and generalize classical equalities and inequalities. Unlike sets and vector spaces, in many categories the rank of an object does not identify it up to isomorphism: to preserve information about the structure of persistence modules, we define colorable ranks, persistence diagrams and prove the equality between multicolored bottleneck distance and interleaving distance in semisimple Abelian categories. To illustrate our framework in practice, we give examples of multicolored persistent homology on filtered topological spaces with a group action and labeled point cloud data.

Motivation & Objective

• To unify topological and combinatorial persistence theories under a common categorical framework using rank functions.
• To generalize classical persistence tools—like persistence diagrams, interleaving, and bottleneck distances—to arbitrary regular and Abelian categories.
• To extend persistent homology to structured data such as filtered spaces with group actions or labeled point clouds by introducing colored ranks and multicolored persistence diagrams.
• To prove that in semisimple Abelian categories, the multicolored bottleneck distance equals the interleaving distance, generalizing classical stability results.
• To provide a foundation for structured data analysis in machine learning by incorporating label information into topological similarity measures.

Proposed method

• Define a ranked category as a regular category equipped with an integer-valued rank function on objects, generalizing dimension and cardinality.
• Construct categorical persistence functions from functors to ranked categories, extending the coherent sampling method from [4] to richer target categories.
• Introduce fiber-wise rank functions as a systematic way to build rank functions, recovering dimension and cardinality as special cases.
• Define persistence diagrams via cornerpoints and their multiplicities in the categorical setting, generalizing the classical construction.
• Introduce coloring of persistence diagrams to preserve structural information (e.g., group action or labels), leading to multicolored bottleneck distance.
• Prove that in semisimple Abelian categories, the interleaving distance equals the multicolored bottleneck distance, under color-preserving bijections.

Experimental results

Research questions

• RQ1 Can a unified categorical framework generalize both topological and combinatorial persistence theories using rank functions?
• RQ2 What axioms must a rank function satisfy to support persistence diagrams and stable distances in arbitrary categories?
• RQ3 How can persistence be extended to structured data such as filtered spaces with group actions or labeled point clouds?
• RQ4 Under what conditions does the multicolored bottleneck distance equal the interleaving distance in generalized persistence frameworks?
• RQ5 Can colored persistence diagrams provide a topological similarity measure that respects additional data structures like labels or symmetries?

Key findings

• The paper establishes a general framework for persistence using rank functions in regular and Abelian categories, unifying classical and combinatorial approaches.
• It proves that in semisimple Abelian categories, the interleaving distance equals the multicolored bottleneck distance when bijections are restricted to preserve colors.
• The framework allows the construction of multicolored persistence diagrams for filtered topological spaces with group actions, preserving group representation structure.
• Labeled point cloud data can be analyzed via multicolored persistence, where labels define the coloring and enable structured similarity measures.
• The generalized persistence framework preserves interleavings and $\epsilon$-interleavings under arbitrary functors, enabling hierarchical analysis across categories.
• The fiber-wise rank construction recovers classical rank functions such as dimension in vector spaces and cardinality in sets as special cases.