Exact weights, path metrics, and algebraic Wasserstein distances

TL;DR

The article develops a unified framework to measure distances between objects in abelian and Grothendieck categories via weights, introducing path metrics that are equivalent to exactness, stability, and lower-bounding properties. It extends these path metrics to generalized persistence modules indexed by measure spaces and constructs Wasserstein distances W_p(d) that extend classical diagram-based distances while preserving a universal property. For one-parameter persistence modules, the work proves isometries between the algebraic Wasserstein distance and the diagrammatic Wasserstein distance, and establishes induced algebraic matchings for maps into and out of interval modules, including a W_1 isometry theorem. The framework then applies to multiparameter and zigzag persistence, illustrating concrete distance computations and highlighting the nuanced differences between path metrics and Wasserstein-type metrics in more complex indexing categories. Overall, the results provide a robust, algebraically grounded toolkit for comparing generalized persistence modules with clear theoretical guarantees and practical implications for multi-parameter and zigzag settings.

Abstract

We use weights on objects in an abelian category to define what we call a path metric. We introduce three special classes of weight: those compatible with short exact sequences; those induced by their path metric; and those which bound their path metric. We prove that these conditions are in fact equivalent, and call such weights exact. As a special case of a path metric, we obtain a distance for generalized persistence modules whose indexing category is a measure space. We use this distance to define Wasserstein distances, which coincide with the previously defined Wasserstein distances for one-parameter persistence modules. For one-parameter persistence modules, we also describe maps to and from an interval module, and we give a matrix reduction for monomorphisms and epimorphisms.

Figures (3)

• Figure 1: A one dimensional simplicial complex $K$ (top) and a pair of two-parameter filtrations, $X$ (bottom left) and $Y$ (bottom right). The differences between $X$ and $Y$ are highlighted.
• Figure 2: A one dimensional simplicial complex $K$ (top) and a pair of two-parameter filtrations, $X_t$ (bottom left) and $X_1$ (bottom right). The difference between $X_t$ and $X_1$ is highlighted.
• Figure 3: Middle: a region in the plane whose boundary is a trapezoid. Left: a subset of this region obtained by removing the triangular subregion on the right.

Theorems & Definitions (122)

• Theorem 1.1: Theorem \ref{['thm:equivalent-conditions']}
• Theorem 1.2: Proposition \ref{['prop:upper-bound-pm']} and Theorem \ref{['thm:lower-bound-pm']}
• Theorem 1.3
• Theorem 1.4: Theorem \ref{['thm:Wp-isometry']}
• Theorem 1.5: Theorem \ref{['thm:metrics-same']}
• Theorem 1.6: Theorem \ref{['thm:matching-mono']}
• Theorem 1.7: Theorems \ref{['thm:matching-epi']}
• Lemma 1.8: Lemma \ref{['lem:interval-nonzero-map']}
• Theorem 1.9: Theorem \ref{['thm:coker']}
• Theorem 1.10: Theorem \ref{['thm:ker']}