内在交织距离用于合并树

通过损失函数在具体范畴上界定交错距离

Bounding the interleaving distance on concrete categories using a loss function

通过引入损失函数，在具体范畴上界定了通用持续模块的交错距离，解决了计算交错距离的NP难问题。

Geometry-Aware Merge Tree Comparisons for Time-Varying Data with Interleaving Distances

Merge trees, a type of topological descriptor, serve to identify and summarize the topological characteristics associated with scalar fields. They present a great potential for the analysis and visualization of time-varying data. First, they give compressed and topology-preserving representations of data instances. Second, their comparisons provide a basis for studying the relations among data instances, such as their distributions, clusters, outliers, and periodicities. A number of comparative measures have been developed for merge trees. However, these measures are often computationally expe…

FPT-algorithms for computing Gromov-Hausdorff and interleaving distances between trees

Gromov-Hausdorff (GH) distance is a natural way to measure the distortion between two metric spaces. However, there has been only limited algorithmic development to compute or approximate this distance. We focus on computing the Gromov-Hausdorff distance between two metric trees. Roughly speaking, a metric tree is a metric space that can be realized by the shortest path metric on a tree. Previously, Agarwal et al. showed that even for trees with unit edge length, it is NP hard to approximate the GH distance between them within a factor of 3. In this paper, we present a fixed-parameter tractabl…

The Universal $\ell^p$-Metric on Merge Trees

Adapting a definition given by Bjerkevik and Lesnick for multiparameter persistence modules, we introduce an $\ell^p$-type extension of the interleaving distance on merge trees. We show that our distance is a metric, and that it upper-bounds the $p$-Wasserstein distance between the associated barcodes. For each $p\in[1,\infty]$, we prove that this distance is stable with respect to cellular sublevel filtrations and that it is the universal (i.e., largest) distance satisfying this stability property. In the $p=\infty$ case, this gives a novel proof of universality for the interleaving distance …

基于带标号合并树交错距离的几何图距离

A Distance for Geometric Graphs via the Labeled Merge Tree Interleaving Distance

针对几何图间距离度量问题，提出基于方向变换旋转子水平集构建合并树、结合满射多标记方案与代表矩阵计算的距离指标，具备理论优良性与两种多项式时间算法，经数据集验证具实用性。

通过路径保持映射计算合并树间交错距离的改进FPT算法

An Improved FPT Algorithm for Computing the Interleaving Distance between Merge Trees via Path-Preserving Maps

本文提出了一种改进的FPT算法，用于计算合并树间的ε-良好映射，通过引入新的参数η_f和η_g，显著降低了计算复杂度。

$\ell^\infty$-Cophenetic度量用于系统发育树作为交错度量

The $\ell^\infty$-Cophenetic Metric for Phylogenetic Trees as an Interleaving Distance

本文研究了系统发育树的$\ell^\infty$-cophenetic度量，并证明其为交错度量的一种实例，通过将系统发育树视为具有额外结构的合并树类别，并利用流的定义来构建交错度量。

A Structural Average of Labeled Merge Trees for Uncertainty Visualization

Physical phenomena in science and engineering are frequently modeled using scalar fields. In scalar field topology, graph-based topological descriptors such as merge trees, contour trees, and Reeb graphs are commonly used to characterize topological changes in the (sub)level sets of scalar fields. One of the biggest challenges and opportunities to advance topology-based visualization is to understand and incorporate uncertainty into such topological descriptors to effectively reason about their underlying data. In this paper, we study a structural average of a set of labeled merge trees and us…

A Distance Between Filtered Spaces Via Tripods

We present a simplified treatment of stability of filtrations on finite spaces. Interestingly, we can lift the stability result for combinatorial filtrations from [CSEM06] to the case when two filtrations live on different spaces without directly invoking the concept of interleaving. We then prove that this distance is intrinsic by constructing explicit geodesics between any pair of filtered spaces. Finally we use this construction to obtain a strengthening of the stability result.

计算$p$-展示距离是困难的

Computing $p$-presentation distances is hard

本文证明了对所有$p\in[1,\infty)$，计算合并树和$t\geq2$参数持久性模块的$p$-展示距离均为NP困难，并提出一种可推广到其他基于求和或$p$-范数距离的新颖证明策略。

Reeb图距离的紧致拟通用性

Tight quasi-universality of Reeb graph distances

本文证明了Reeb图间多种距离的紧致双Lipschitz界以确立其拟通用性，提出了新的函数扭曲距离，并证明了轮廓树及合并树特例下的严格通用性与距离一致性。

Local Equivalence and Intrinsic Metrics between Reeb Graphs

As graphical summaries for topological spaces and maps, Reeb graphs are common objects in the computer graphics or topological data analysis literature. Defining good metrics between these objects has become an important question for applications, where it matters to quantify the extent by which two given Reeb graphs differ. Recent contributions emphasize this aspect, proposing novel distances such as {\em functional distortion} or {\em interleaving} that are provably more discriminative than the so-called {\em bottleneck distance}, being true metrics whereas the latter is only a pseudo-metric…

使用局部敏感哈希的合并树快速比较分析

Fast Comparative Analysis of Merge Trees Using Locality Sensitive Hashing

针对现有合并树相似性度量处理大规模数据扩展性差的问题，提出基于局部敏感哈希的比较框架，结合递归最小哈希和子路径签名扩展实现两种高效相似性度量，效果接近现有方法且可应用于多类场景。

Frechet-Like Distances between Two Merge Trees

The purpose of this paper is to extend the definition of Frechet distance which measures the distance between two curves to a distance (Frechet-Like distance) which measures the similarity between two rooted trees. The definition of Frechet-Like distance is as follows: Tow men start from the roots of two trees. When they reach to a node with the degree of more than $2$, they construct $k-1$ men which $k$ is the outgoing degree of the node and each man monitor a man in another tree (there is a rope between them). The distance is the minimum length of the ropes between the men and the men whom a…

基于部分最优传输的灵活概率拓扑追踪

Flexible and Probabilistic Topology Tracking with Partial Optimal Transport

本文提出一种基于合并树与部分最优传输的灵活概率框架，通过测度网络建模和部分匹配实现时变标量场中拓扑特征的追踪，并验证了其稳定性与有效性。