Related papers: Categorified Reeb Graphs
The Reeb space of a generic map is the space of all connected components of preimages of the map. Reeb spaces are fundamental and useful tools in the theory of Morse functions and higher dimensional variants and their applications to…
Constructing Morse functions and their higher dimensional versions or fold maps is fundamental, important and challenging in investigating the topologies and the differentiable structures of differentiable manifolds via Morse functions,…
A metric graph is a 1-dimensional stratified metric space consisting of vertices and edges or loops glued together. Metric graphs can be naturally used to represent and model data that take the form of noisy filamentary structures, such as…
Inspired by the interval decomposition of persistence modules and the extended Newick format of phylogenetic networks, we show that, inside the larger category of \textit{ordered Reeb graphs}, every Reeb graph with $n$ leaves and first…
We initiate the study of sheaves on Cech closure spaces, providing a new, unified approach to sheaf theory on many of the major classes of spaces of interest to applications: topological spaces, finite simplicial complexes (seen as $T_0$…
A graph-based classification method is proposed for semi-supervised learning in the case of Euclidean data and for classification in the case of graph data. Our manifold learning technique is based on a convex optimization problem involving…
Graph structure learning aims to learn connectivity in a graph from data. It is particularly important for many computer vision related tasks since no explicit graph structure is available for images for most cases. A natural way to…
In this work we are focused on the existence of Morse functions on a closed manifold $M$ which are far from being ordered, i.e. whose Reeb graphs have positive first Betti number, especially the maximal possible, equals…
Data consisting of a graph with a function mapping into $\mathbb{R}^d$ arise in many data applications, encompassing structures such as Reeb graphs, geometric graphs, and knot embeddings. As such, the ability to compare and cluster such…
We present an exact and efficient algorithm for computing the Reeb space of a bivariate PL map. The Reeb space is a topological structure that generalizes the Reeb graph to the setting of multiple scalar-valued functions defined over a…
Mapper graphs are widely used tools in topological data analysis and visualization. They can be understood as discrete approximations of Reeb graphs, providing insight into the shape and connectivity of complex data. Given a…
In the singularity and differential topological theory of Morse functions and higher dimensional versions or fold maps and application to algebraic and differential topology of manifolds, constructing explicit fold maps and investigating…
Simple polyhedra are $2$-dimensional polyhedra and important objects in low-dimensional geometry and in the applications of {\it fold} maps, defined as smooth maps regarded as higher dimensional variants of Morse functions. For example,…
We introduce decorated mapper graphs as a generalization of mapper graphs capable of capturing more topological information of a data set. A decorated mapper graph can be viewed as a discrete approximation of the cellular Leray cosheaf over…
Searching topological similarity between a pair of shapes or data is an important problem in data analysis and visualization. The problem of computing similarity measures using scalar topology has been studied extensively and proven useful…
In our paper, we introduce special-generic-like maps or SGL maps as smooth maps and study their several algebraic topological and differential topological properties. The new class generalize the class of so-called special generic maps.…
The high-level structure of a graph is a crucial ingredient for the analysis and visualization of relational data. However, discovering the salient graph patterns that form this structure is notoriously difficult for two reasons. (1)…
Discrete Morse theory emerged as an essential tool for computational geometry and topology. Its core structures are discrete gradient fields, defined as acyclic matchings on a complex $C$, from which topological and geometrical informations…
Temporal graphs are commonly used to represent time-resolved relations between entities in many natural and artificial systems. Many techniques were devised to investigate the evolution of temporal graphs by comparing their state at…
This article proposes to integrate two Reeb graphs with the information of their isosurfaces' inclusion relation. As computing power evolves, there arise numerical data that have small-scale physics inside larger ones -- for example, small…