A family of metrics from the truncated smoothing of Reeb graphs

TL;DR

This paper introduces a new family of metrics for Reeb graphs based on truncated smoothing, which generalizes the interleaving distance and offers stability properties, enhancing the analysis of topological data structures.

Contribution

The authors define a novel truncated smoothing technique for Reeb graphs, leading to a family of generalized metrics that extend the interleaving distance with stability properties.

Findings

Defined a truncated smoothing functor for Reeb graphs.

Constructed a categorical flow and interleaving distance for the new metrics.

Showed metrics are strongly equivalent and stable up to a constant.

Abstract

In this paper, we introduce an extension of smoothing on Reeb graphs, which we call truncated smoothing; this in turn allows us to define a new family of metrics which generalize the interleaving distance for Reeb graphs. Intuitively, we "chop off" parts near local minima and maxima during the course of smoothing, where the amount cut is controlled by a parameter $\tau$. After formalizing truncation as a functor, we show that when applied after the smoothing functor, this prevents extensive expansion of the range of the function, and yields particularly nice properties (such as maintaining connectivity) when combined with smoothing for $0 \leq \tau \leq 2\varepsilon$, where $\varepsilon$ is the smoothing parameter. Then, for the restriction of $\tau \in [0,\varepsilon]$, we have additional structure which we can take advantage of to construct a categorical flow for any choice of slope $m \in [0,1]$. Using the infrastructure built for a category with a flow, this then gives an interleaving distance for every $m \in [0,1]$, which is a generalization of the original interleaving distance, which is the case $m=0$. While the resulting metrics are not stable, we show that any pair of these for $m,m' \in [0,1)$ are strongly equivalent metrics, which in turn gives stability of each metric up to a multiplicative constant. We conclude by discussing implications of this metric within the broader family of metrics for Reeb graphs.