Towards Computing Average Merge Tree Based on the Interleaving Distance

TL;DR

An average merge tree for a pair of merge trees is defined using the interleaving distance and it is proved that the resulting merge tree indeed satisfies a natural notion of averaging for the two given merge trees.

Abstract

The interleaving distance is a key tool for comparing merge trees, which provide topological summaries of scalar functions. In this work, we define an average merge tree for a pair of merge trees using the interleaving distance. Since such an average is not unique, we propose a method to construct a representative average merge tree. We further prove that the resulting merge tree indeed satisfies a natural notion of averaging for the two given merge trees. To demonstrate the structure of the average merge tree, we include illustrative examples.

Figures (5)

• Figure 1: Left — A comparison of two mappings $\alpha$ and $\beta$, where vertex $u$ is mapped to $\alpha(u)$, located ${\varepsilon}$ above $u$, and $\beta(\alpha(u))$ is an ancestor of $u$, situated $2{\varepsilon}$ higher; Right — An example where the maps $\alpha$ and $\beta$ are not ${\varepsilon}$-interleaved, since $\beta(\alpha(a)) \neq a^{2{\varepsilon}}$.
• Figure 2: Illustrations of interleaving distance differences, ${u'}_1$ and $u_2$ map to the same point, but their $2{\varepsilon}$-extensions differ: ${u'}_1^{2{\varepsilon}} \neq u_2^{2{\varepsilon}}$.
• Figure 3: Two merge trees ${T_1^f}$ and ${T_2^g}$ are displayed on the left. The process consists of the following steps: (Step I) construct the augmented trees ${\hat{T}_1^f}$ and ${\hat{T}_2^g}$ and identify the ${\varepsilon}$-good map $\alpha$. (Step II) shift ${\hat{T}_1^f}$ upward by $\frac{{\varepsilon}}{2}$. (Step III) incorporate edges from ${\hat{T}_2^g}\setminus Img(\alpha)$; and (Step IV) for nodes that are mapped to single point in ${\hat{T}_2^g}$ shift their nearest common ancestor downward by ${\varepsilon}$.
• Figure 4: ${\varepsilon}$-ball around vertex $u$. The ${\varepsilon}$-degree around $u$ is $10$ which is the summation of all the nodes in the ${\varepsilon}$-ball.
• Figure 5: Top: the left and right trees show the first two leaf datasets from MiSoTS. Bottom: the corresponding input merge trees $T_1$ and $T_2$, with the associated colorbar and their average merge tree. We display only the labels of the leaf nodes.

Theorems & Definitions (16)

• definition 1: FAfCGHaIDbT
• definition 2: IDbMT
• definition 3: FAfCGHaIDbT
• theorem 1: FAfCGHaIDbT
• theorem 2
• proof
• theorem 3
• proof
• Claim 1
• proof