Bounding the interleaving distance on concrete categories using a loss function

TL;DR

This work introduces a loss-based framework for bounding the interleaving distance between generalized persistence modules valued in concrete categories. By defining a merging distance on module elements and a loss function for assignments, the authors prove the bound $d_I(F,G) \le \varepsilon + L(\varphi,\psi)$ and develop polynomial-time algorithms in FC categories and in $\mathrm{Vec}_\mathbb{F}$ under certain conditions. The method generalizes previous mapper-graph approaches to broader inputs and provides scalable computation through reductions over linear and product posets, as well as constructive assignments. The results offer a practical route to obtaining upper bounds on interleaving distance for multiparameter persistence and related structures, with potential extensions to non-concrete categories and more general interleaving notions.

Abstract

The interleaving distance is arguably the most widely used metric in topological data analysis (TDA) due to its applicability to a wide array of inputs of interest, such as (multiparameter) persistence modules, Reeb graphs, merge trees, and zigzag modules. However, computation of the interleaving distance in the vast majority of this settings is known to be NP-hard, limiting its use in practical settings. Inspired by the work of Chambers et al. on the interleaving distance for mapper graphs, we solve a more general problem bounding the interleaving distance between generalized persistence modules on concrete categories via a loss function. This loss function measures how far an assignment, which can be thought of as an interleaving that might not commute, is from defining a true interleaving. We give settings for which the loss can be computed in polynomial time, including for certain assumptions on $k$-parameter persistence modules.

Figures (6)

• Figure 1: A index category $J$ and square diagram on a category $C$ with objects $W,X,Y,Z$ and morphisms $f,g,h,i$
• Figure 2: Example of the construction of the intervals $I_{i,k}$ for the map $\mathcal{T}_1: \mathbb{R} \to \mathbb{R}$ given by $\mathcal{T}_1(m) = \lfloor m\rfloor + 1$ and the functors $F: \mathbb{R} \to C$ with critical set $\{c_1,c_2,c_3\}$ and $G: \mathbb{R} \to C$ with critical set $\{d_1,d_2,d_3\}$. For each interval we have $\alpha_{i,k} = I_{i,k}$.
• Figure 3: On the left: Example of the construction of the cubes $S_{(j_1,j_2)} = X^1_{j_1} \times X^2_{j_2}$ for the functors $F: P^2 \to C$ with critical coordinates $\{x^1_2,x^1_4\}$ and $\{x^2_2,x^2_4\}$. On the right: We divide the cube $S_{33}$ into interval $\mathcal{I}_J^{33}$ that map into the constant cube $R_J$ by the morphism $\mathcal{T}_\varepsilon$. For example, $p \in \mathcal{I}_{12}^{33}$ if and only if $p \in S_{33}$ and $\mathcal{T}_\varepsilon p \in R_{12}$.
• Figure 4: Diagrams for defining $\hat{F}$ in the proof of Thm. \ref{['thm:hatF_is_functor']}.
• Figure 5: Diagrams that show the existence of the maps $\hat{F}(\alpha_{J}) \to \hat{G}(\mathcal{T}_\varepsilon \alpha_{J})$ and $\hat{G}(\alpha_{J}) \to \hat{F}( \mathcal{T}_\varepsilon\alpha_{J})$ by the universal property.

Theorems & Definitions (82)

• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3: Thin lemma
• Corollary 2.4
• Corollary 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Lemma 2.9