[Paper Review] Wasserstein distance for generalized persistence modules and abelian categories

Peter Bubenik, Jonathan Scott+1 more|arXiv (Cornell University)|Sep 25, 2018

Topological and Geometric Data Analysis18 references3 citations

TL;DR

This paper introduces an algebraic formulation of Wasserstein distances for generalized persistence modules, extending the standard L^p distances beyond persistence diagrams to all persistence modules. For p=1, the definition generalizes to abelian categories, and for arbitrary p, to Krull-Schmidt categories, enabling Wasserstein distances for multi-parameter persistence modules and offering a foundation for their computation.

ABSTRACT

In persistence theory and practice, measuring distances between modules is central. The Wasserstein distances are the standard family of L^p distances for persistence modules. They are defined in a combinatorial way for discrete invariants called persistence diagrams that are defined for certain persistence modules. We give an algebraic formulation of these distances that applies to all persistence modules. Furthermore, for p=1 this definition generalizes to abelian categories and for arbitrary p it generalizes to Krull-Schmidt categories. In particular, we obtain a definition of Wasserstein distance for multi-parameter persistence modules. These distances may be useful for the computation of distance between generalized persistence modules. In our most technical proof, we classify certain maps of persistence modules, which may be of independent interest.

Motivation & Objective

• To provide a unified, algebraic formulation of Wasserstein distances applicable to all persistence modules, not just those with persistence diagrams.
• To extend the definition of Wasserstein distances to abelian categories for p=1 and to Krull-Schmidt categories for arbitrary p.
• To enable the use of Wasserstein distances in multi-parameter persistence modules, where traditional diagram-based definitions fail.
• To support the computational analysis of generalized persistence modules through a robust, category-theoretic distance framework.

Proposed method

• Develop an algebraic definition of Wasserstein distances using the structure of persistence modules and their decompositions.
• Leverage the Krull-Schmidt theorem to define distances via direct sum decompositions in Krull-Schmidt categories.
• Apply category-theoretic tools to generalize the L^p distance to abelian categories when p=1.
• Use combinatorial maps between modules to define optimal matchings of indecomposable components, forming the basis of the distance.
• Establish a correspondence between optimal matchings and minimal resolutions in the module category.
• Prove a technical classification of module maps that underpins the distance construction and may be of independent interest.

Experimental results

Research questions

• RQ1 How can Wasserstein distances be defined algebraically for persistence modules without relying on persistence diagrams?
• RQ2 Can the Wasserstein distance be generalized from discrete diagrams to the full category of persistence modules?
• RQ3 Does the p=1 Wasserstein distance extend naturally to abelian categories beyond the standard persistence module setting?
• RQ4 Can the framework be extended to multi-parameter persistence modules through a categorical generalization?
• RQ5 What structural properties of module maps enable the construction of optimal matchings for distance computation?

Key findings

• The paper provides a fully algebraic definition of Wasserstein distances that applies to all persistence modules, not just those with persistence diagrams.
• For p=1, the Wasserstein distance generalizes to abelian categories, enabling its use in broader categorical settings.
• For arbitrary p, the definition extends to Krull-Schmidt categories, which include multi-parameter persistence modules.
• The construction relies on a classification of module maps that ensures the existence of optimal matchings between indecomposable components.
• The framework supports the computation of distances in multi-parameter persistence, where diagram-based methods are not generally applicable.
• The technical classification of maps in the proof may have independent applications in module theory and representation theory.