Related papers: Computing a Stable Distance on Merge Trees

Frequencies of $k$-mers in sequences are sometimes used as a basis for inferring phylogenetic trees without first obtaining a multiple sequence alignment. We show that a standard approach of using the squared-Euclidean distance between…

Different types of two- and three-dimensional representations of a finite metric space are studied that focus on the accurate representation of the linear order among the distances rather than their actual values. Lower and upper bounds for…

Quantifying the distance between datasets is a fundamental question in mathematics and machine learning. We propose \textit{magnitude distance}, a novel distance metric defined on finite datasets using the notion of the \emph{magnitude} of…

Motivated from the study of eccentricity, center, and sum of eccentricities in graphs and trees, we introduce several new distance-based global and local functions based on the smallest distance from a vertex to some leaf (called the…

Large graphs are difficult to represent, visualize, and understand. In this paper, we introduce "gate graph" - a new approach to perform graph simplification. A gate graph provides a simplified topological view of the original graph.…

Similarity searching finds application in a wide variety of domains including multilingual databases, computational biology, pattern recognition and text retrieval. Similarity is measured in terms of a distance function, edit distance, in…

Merge trees are a powerful tool from topological data analysis that is frequently used to analyze scalar fields. The similarity between two merge trees can be captured by an interleaving: a pair of maps between the trees that jointly…

We propose a novel technique, termed compact shape trees, for computing correspondences of single-boundary 2-D shapes in O(n2) time. Together with zero or more features defined at each of n sample points on the shape's boundary, the compact…

We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence. Two functions are considered persistence equivalent if and only if they induce the same persistence diagram. We compare this…

The matrices of spanning rooted forests are studied as a tool for analysing the structure of digraphs and measuring their characteristics. The problems of revealing the basis bicomponents, measuring vertex proximity, and ranking from…

Among subgraphs with a fixed number of vertices of the regular square lattice, we prove inequalities that essentially say that those with smaller boundaries have larger numbers of spanning trees and vice-versa. As an application, we relate…

Common measures of neural representational (dis)similarity are designed to be insensitive to rotations and reflections of the neural activation space. Motivated by the premise that the tuning of individual units may be important, there has…

Node similarity is a fundamental problem in graph analytics. However, node similarity between nodes in different graphs (inter-graph nodes) has not received a lot of attention yet. The inter-graph node similarity is important in learning a…

Tree congruence metrics are typically global indices that describe the similarity or dissimilarity between dendrograms. This study principally focuses on topological congruence metrics that quantify similarity between two dendrograms and…

Alternative novel measures of the distance between any two partitions of a n-set are proposed and compared, together with a main existing one, namely 'partition-distance' D(.,.). The comparison achieves by checking their restriction to…

Identifying and comparing topological features, particularly cycles, across different topological objects remains a fundamental challenge in persistent homology and topological data analysis. This work introduces a novel framework for…

This paper introduces \textit{measurement trees}, a novel class of metrics designed to combine various constructs into an interpretable multi-level representation of a measurand. Unlike conventional metrics that yield single values,…

Mathematical modeling is a standard approach to solve many real-world problems and {\em diversity} of solutions is an important issue, emerging in applying solutions obtained from mathematical models to real-world problems. Many studies…

Geometric characteristics of metric spaces that appear in formulas of the Gromov--Hausdorff distances from these spaces to so-called simplexes, i.e., to the metric spaces, all whose non-zero distances are the same are studied. The…

There is a rising interest in mapping trees using satellite or aerial imagery, but there is no standardized evaluation protocol for comparing and enhancing methods. In dense canopy areas, the high variability of tree sizes and their spatial…